Large Cardinals and Elementary Embeddings of V

Large cardinals and elementary embeddings of V

Master thesis of: Marios Koulakis

Supervisor: Athanassios Tzouvaras,

Professor

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To the memory of my parents, Ioannis Koulakis and Anastasia Chrysochoidou

Contents

1 Preliminaries 1 1.1 Cumulative hierarchies ...... 1 1.2 TheMostowskicollapse ...... 5 1.3 Absoluteness ...... 6 1.4 Vα and Lα ...... 12 1.5 Modeltheory ...... 17

2 Some smaller large cardinals 21 2.1 Cardinalarithmetic...... 22 2.2 Inaccessiblecardinals...... 26 2.3 Mahlocardinals...... 30 2.4 Weakly compact,Erd˝osand Ramsey cardinals...... 33

3 Measurable cardinals and elementary embeddings of V 37 3.1 Aspectsofmeasurability...... 37 3.2 Elementary embeddings of V ...... 42 3.3 Normal measures, indescribability ...... 48

4 0♯ and elementary embeddings of L 53 4.1 Indiscernibles ...... 53 4.2 0♯ ...... 54 4.3 The connection of 0♯ with L ...... 60 4.4 Elementary embeddings of L ...... 62

5 Stronger embeddings of V 69 5.1 Strong,Woodinandsuperstrongcardinals ...... 69 5.2 Stronglycompactcardinals ...... 72 5.3 Supercompactcardinals ...... 73 5.4 Extendible cardinals and Vop˘enka’s principle ...... 75 5.5 HugecardinalsandI0-I3...... 76 5.6 Kunen’stheorem ...... 78 5.7 Thewholenessaxiom...... 79

i ii CONTENTS Introduction

This thesis is a short survey of the theory of large cardinals. The notion of elementary embeddings is crucial in this study, since it is used to provide a quite general way of defining large cardinals. The survey concludes with a reference to Kunen’s theorem, which sets some limitations to the existence of specific elementary embeddings, and an attempt of P. Corazza to surmount those limitations. Most of the material used through the survey is defined in chapter 1, thus the text is accessible to all readers having knowledge of basic set theory and logic. Chapter 1 serves as an extensive introduction to some notions needed for the rest of the text. Mostowski’s collapsing theorem provides us with a way of transforming a lot of models of set theory, into standard transitive ones. Some results concerning Lévi’s hierarchy illuminate the notion of absoluteness, i.e. the case in which a formula or a term behaves the same way in V as in a standard transitive model. The important cumulative hierarchies Vα and Lα are introduced, along with some of their basic properties. Finally, there is a quick reference to some model theoretical constructions we will use later on. On chapter 2, the first large cardinals appear, though they are not strong enough to yield a connection with elementary embeddings. The first section of this chapter is somehow a continuation of the first chapter, as it provides some fundamental knowledge on cardinal arithmetic. Afterwards, we define inaccessible cardinals, then go on to Mahlo cardinals and finally introduce weakly compact and Erd˝os cardinals. There is an additional reference to Ramsey cardinals, which have a combinatorial nature similar to that of Erd˝os cardinals, even though they are essentially stronger than the other large cardinals we have mentioned here. On chapter 3 things seem to get more interesting, as we introduce measurable cardinals. They emerge in a very natural way from measure theory and their study moves from analysis to set theory. They are connected with the elementary embeddings of the form j : V ≺ M, and out of this connection we get the result, due to Scott, that if there is a measurable cardinal, then V 6= L. Concluding, we define normal measures, which become invaluable in the study of elementary embeddings. Chapter 4 goes a little bit backwards, since we study a weaker large cardinal hypothesis, the existence of 0♯. Some effort is needed in order to define 0♯ but it pays back, as 0♯ is very close to the critical question of whether L is a good

iii iv CONTENTS approach of V . Silver’s theorem and Jensen’s covering theorem give a more than satisfactory answer to this question which had been roughly approached by measuble and Ramsey cardinals. We ﬁnally see, through a theorem of Kunen, that 0♯ is connected with the elementary embeddings of the form j : L ≺ L. Chapter 5 goes on from the point where chapter 3 ended, as it presents the use of elementary embeddings of the form j : V ≺ M, in order to obtain stronger large cardinals. This way, strong, Woodin, strongly compact, supercompact and extendible cardinals appear. We come up with even stronger hypotheses, such as Vop˘enka’s principle, the existence of huge cardinals and I0-I3. The latter, seem to lie near the existence of an embedding j : V ≺ V , which cannot exist due to Kunen’s result, thus there is a probability that they could be disproved from ZF C. Finally, we examine the case of an elementary embedding j : V ≺ V . Kunen’s theorem states that there is not such an embedding. In the process, we examine the consequences of the wholeness axiom, proposed by Corazza, which asserts the existence of an embedding j : V ≺ V that is not weakly deﬁnable. The wholeness axiom may weaken the hypotheses of Kunen’s result, but it implies the existence of a super-n-huge cardinal, thus it is very near inconsistency and it should be further investigated. Chapter 1

Preliminaries

In this chapter, we present some basic notions of set theory and model theory which will be necessary for our study. On the same time, we try to give some motives for the introduction of large cardinals. Our first step is to present the well known axiomatic system ZF C. Most of the axioms are presented as the closure of V under certain operations, in order to provide a more vivid image of the structure of V . Afterwards, the most powerful of those operations, the powerset operation, is used in order to introduce the notion of cumulative hierarchies. Cumulative hierarchies are roughly sequences of levels Uα, which expand, by the addition of more complex sets, as α gets bigger. Two very common hierarchies are then introduced; von Newmann’s hierarchy Vα, and the hierarchy of the constructible sets Lα. Moving towards a model theoretical study of set theory, Mostowski’s collapsing theorem becomes very useful, as it gives us the opportunity to work, in almost all cases studied here, with standard transitive models. Since standard models share the same relation with V , we are interested in the cases where some formulas have the same truth value in a stander transitive model as in V . The notion that expresses those cases is absoluteness, and an easy way to study it is the introduction of Lévi’s hierarchy of formulas. Next, we meet again the hierarchies Vα, Lα and this time prove some of their basic properties. Finally, we state some model theoretic concepts, such as Skolem functions, ultraproducts and reflection principles. For a more extensive treatment of the material covered in this chapter, the reader is referred to [7], [11] and [21]. For an elementary introduction to set theory, he is referred to [15] and [12]. A classical book on model theory is [2] and much more information on ultraproducts can be found in [1].

1.1 Cumulative hierarchies

The axioms of ZFC The ﬁrst two axioms describe the nature of sets:

1 2 CHAPTER 1. PRELIMINARIES

• A1. Extensionality axiom

∀z (z ∈ x ↔ z ∈ y) → x = y

• A2. Foundation axiom

∃y y ∈ x → ∃y ∈ x ∀z ∈ xz∈ / y

The next 7 axioms express that the universe V is closed under certain operations, and are written in the form

x ∈ V → Q(x) ∈ V where Q denotes an operation. These are the following:

• A3φ. Separation axiom

x ∈ V →{y ∈ x : φ(y)} ∈ V

• A4. Empty set axiom → ∅ ∈ V

• A5. Pairing axiom

x, y ∈ V →{x, y} ∈ V

• A6. Power set axiom

x ∈ V → P (x) ∈ V

• A7. Union axiom x ∈ V → ∪x ∈ V

• A8. Inﬁnity axiom (inf)

∅ ∈ V → ω ∈ V

• A9φ. Replacement axiom (F)

′′ x ∈ V → Fφ (x) ∈ V

′′ Where Fφ (x)= y ↔ ∀z (z ∈ y ↔ ∃w ∈ x φ(w,z)) for φ which deﬁne functions, i.e. ∀x ∃!y φ(x, y). Finally, the tenth axiom, the axiom of choice, demands the existence of a well-ordering for every set. • A10. Choice axiom (AC)

∀x ∃y y wellorders x 1.1. CUMULATIVE HIERARCHIES 3

An immediate consequence of the axiom of foundation is the fact that there can not exist inﬁnite ∋ sequences. If we combine this with the axiom of extensionality, which states that every set is deﬁned exactly by its elements, we can imagine a construction of V , beginning with the empty set and recursively moving to new sets using axioms A4-A10. We could also add new sets which do not appear in this construction, but may be contained in sets we have already been constructed. We should note here that by ZF C − Ai we mean the axiom system ZF C without the axiom Ai, where 1 ≤ i ≤ 10. We also denote, ZF = ZF C − AC and Z = ZF − F .

Cumulative hierarchies The above approach is quite vivid and reveals the structure that V should have, in order to satisfy ZFC. Unfortunately, it is quite complicated, especially when dealing with sets of inﬁnite cardinality. What could be done to simplify it, is to begin with the empty set and produce an ⊂ sequence hUα : α ∈ Oni of sets with increasing complexity, relative to ∈. Using this method, we approach V by constructing bigger and bigger sets of its elements instead of adding them one by one. This idea is captured by the notion of cumulative hierarchies.

Deﬁnition 1.1. A transﬁnite sequence hUα : α ∈ Oni is called a cumulative hierarchy if the following are true:

(i) U0 = ∅;

(ii) Uα ⊂ Uα+1 ⊂ P (Uα);

(iii) Uα = β<α Uβ, limit(α). S The sets Uα of the hierarchy are all transitive. Reaching the end of this section, we are going to introduce two important cumulative hierarchies and some of their trivial properties. More information about them will be unveiled in section 1.4 where we are going to treat them as models of ZFC. The fact that they are indeed cumulative hierarchies can be easily checked in all cases using transﬁnite induction.

Von Neumann’s hierarchy Deﬁnition 1.2. We call von Neumann’s hierarchy the transﬁnite sequence hVα : α ∈ Oni with the following properties:

(i) V0 = ∅;

(ii) Vα+1 = P (Vα);

(iii) Vα = β<α Vβ , limit(α). S Von Neumann’s universe is the class R = {Vα : α ∈ On}. S 4 CHAPTER 1. PRELIMINARIES

• R = V . The power set operation turns out to be so powerful that in ZF every set x belongs to Vα+1 for some α ∈ On. The least α satisfying this property is called the rank of x and is written as rank(x) (We will use the symbolization rankU (x) for the same notion concerning another hierarchy U).

• For every ordinal α, rank(α)= α.

1 • |Vω+α| = iα and for every n ∈ ω, Vn is ﬁnite .

The hierarchy of constructible sets In order to introduce constructible sets, we first need the notion of definability. Definition 1.3.

(i) Suppose M is a model and X, I ⊂ M. X is deﬁnable in M from I, if there is a formula φ(v, u¯) and a n-tuplea ¯ of I, such that:

∀x ∈ M (x ∈ X ↔ M φ[x, a¯]).

In the case I = M we say that X is definable over M. If x ∈ M, then x is definable over M (or in M from I) if {x} is respectively definable over M (or in M from I).

(ii) defM (I)= {X ⊂ M : X definable in M from I} def(M)= {X ⊂ M : X definable over M}.

Deﬁnition 1.4. The hierarchy hLα : α ∈ Oni of the constructible sets is the one which satisﬁes the following:

(i) L0 = ∅;

(ii) Lα+1 = def(Lα);

(iii) Lα = β<α Lβ, limit(α). S The universe of constructible sets is L = {Lα : α ∈ On}. S • α∈On Lα does not necessarily contain all sets. Moreover it cannot be Sproved within ZF C weather V = L or not. An interesting result to come is that this question is tightly connected with the existence of 0♯, which is a large cardinal property.

1 The sequence iα is recursively deﬁned as follows:

i0 = ℵ0

iα iα+1 = 2 i i α = [ β , limit(α) β<α 1.2. THE MOSTOWSKI COLLAPSE 5

• For every ordinal α, rankL(α)= α.

• |Lα| = |α|.

• Lω = Vω and Ln = Vn for every n ∈ ω.

1.2 The Mostowski collapse

In the previous section we introduced cumulative hierarchies in order to get a grasp of how a model of ZF C should look like. In general, a model of set theory is a couple (M, E) where M is a class and E is a binary relation interpreting ∈. The models of ZF C may be quite complicated but we can reduce many of them to simpler ones using Mostowski’s theorem. This theorem states that, under certain assumptions, an E structure is isomorphic to a transitive ∈ class. We give a definition concerning relations and afterwards the proof of Mostowski’s theorem. The reason for this is to illustrate the use of transfinite recursion and provide a clearer image of the connection of some E relations to ∈. Definition 1.5. Suppose (N, E) is a structure of the language {∈}.

(i) The E-extension of x ∈ N is the class extE(x)= {y ∈ N : yEx}. (ii) E is well-founded on N if every non empty subset of N has an E-minimal element.

(iii) E is set-like if for all x ∈ N, extE(x) is a set. (iv) E is extensional relation on N if

∀x, y ∈ N (extE(x)= extE(y) → x = y)

Theorem 1.6. (Mostowski’s collapsing theorem) (i) If E is well-founded set-like and extensional on N, then there exists a transitive class M and an isomorphism F between (N, E) and (M, ∈). (ii) M and F are unique. Proof. (i) We deﬁne F by well-founded recursion so that

′′ F (x)= F (extE (x)) = {F (y): yEx}.

This is possible because E is set-like so extE (x) is a set, which means by the axiom of replacement that {F (y): yEx} is also a set, and additionally E is well-founded. Let M = F ′′(N). – M is transitive: y′ ∈ x′ ∈ M → ∃x ∈ N x′ = F (x) and F (x)= {F (z): zEx} → ∃yEx y′ = F (y) → y′ ∈ M. 6 CHAPTER 1. PRELIMINARIES

– F is 1-1: If it wasn’t, there would exist an x′, of least rank, with the property x′ = F (x)= F (y) for some y 6= x.

y 6= x → extE(x) 6= extE(y) → ∃zEx ¬zEy

but since F (x)= F (y)

∃uEy F (z)= F (u) → z = u → zEy

which is a contradiction. ′′ – yEx ↔ F (y) ∈ F (x): If yEx, then F (y) ∈ F (extE(x)) = F (x). Additionally, if F (y) ∈ F (x), then ∃z ∈ extE(x) F (z) = F (y) and since F is 1-1 we have z = y which means that yEx. (ii) Let F be the above isomorphism and G another one having K as its range. If x ∈ N is an element of least rank for which F (x) 6= G(x), then since G is an isomorphism

G(x)= {G(y): yEx} = {F (y): yEx} = F (x)

which can not be true. As a consequence of this

F = G → K = G′′N = F ′′N = M

so K = M.

In the cases we are interested in, (M, E) will be a model of the axiom of extensionality. Using the following lemma we will show that if this is true, then it is enough to check that extE(x) is a set for every x ∈ M and that E is well-founded, in order to apply Mostowski’s theorem. Lemma 1.7. E is extensional on M iﬀ (M, E) “extensionality axiom”. Proof. M “extensionality axiom” is equivalent to

M ∀x, y (∀z ∈ x z ∈ y ∧ ∀z ∈ y z ∈ x → x = y) ↔ ↔∀x, y ∈ M (∀z ∈ x z ∈ y ∧ ∀z ∈ y z ∈ x → x = y)M ↔∀x, y ∈ M (∀zEx zEy ∧ ∀zEy zEx → x = y) which is equivalent to the extensionality of M.

1.3 Absoluteness

As we have mentioned, we will only work with transitive models (M, ∈), where M can either be a set or a class2. Since the interpretation of ∈ coincides with ∈,

2Here one should be careful using the relation , which is deﬁnable in ZF C for models that are sets, but not for models that are classes. 1.3. ABSOLUTENESS 7 it would be interesting to study the cases where (φ)M is equivalent to φ for all valuations of φ. Those formulas are called absolute for M. If moreover, one of them is absolute for every transitive model M, it is simply called absolute. Our aim in this section is to present methods of proving that a formula is absolute, and using them provide a list of absolute formulas.

Absoluteness Deﬁnition 1.8. Let M and N be two structures such that M ⊂ N: (i) A formula φ(¯x) is preserved under the extension (restriction) from M to N (from N to M) if

∀a¯ ∈ M n(M φ[¯a] → N φ[¯a])

[∀a¯ ∈ M n(N φ[¯a] → M φ[¯a])].

(ii) A fomula φ(¯x) is absolute for M, N if it is preserved both under the extension from M to N and the restriction from N to M. (iii) A term t(¯x) is absolute for M, N if y = t(¯x) is absolute for M, N. (iv) M is an elementary substructure of N, M ≺ N, if every formula is absolute for M, N. (v) We will call a formula φ absolute for a theory T if, for every model M of T , φ is absolute for M, V .

Lévy’s hierarchy One way we can show that a formula is absolute is, by checking its position in Lévy’s hierarchy. We will give a brief definition of it and a theorem which explains the link between absoluteness and the first levels of this hierarchy. Definition 1.9. Let T be a theory. Lévy’s hierarchy and Lévy’s hierarchy relative to T are defined as follows:

(i) – ∆0 =Σ0 = Π0 is the set of all formulas with bounded quantiﬁers (of the form ∃x ∈ y or ∀x ∈ y).

– Σn+1 = {∃x φ(x): φ ∈ Πn}.

– Πn+1 = {∀x φ(x): φ ∈ Σn}.

T (ii) – φ ∈ Σn iff ∃ψ ∈ Σn T ⊢ φ ↔ ψ. T – φ ∈ Πn iff ∃ψ ∈ Πn T ⊢ φ ↔ ψ. T T T – ∆n =Σn ∩ Πn . Theorem 1.10. Let M ⊂ N be transitive models of T , then: (i) If T ⊢ φ ↔ ψ then ψ is absolute for M, N iff φ is absolute for M, N. 8 CHAPTER 1. PRELIMINARIES

T (ii) Every ∆0 sentence is absolute for M, N.

T (iii) Every Σ1 sentence is preserved under the extension from M to N.

T (iv) Every Π1 sentence is preserved under the restriction from N to M.

T (v) Every ∆1 sentence is absolute for M, N. In order to use the above theorem eﬀectively, the lemma below is very useful. It helps us classify formulas in L´evy’s hierarchy.

Lemma 1.11.

(i) Logical connectives (for T ⊢ Z − inf)

T T – If φ and ψ are both Σn or Πn , then so are φ ∧ ψ and φ ∨ ψ. T T T T – If φ ∈ Σn and ψ ∈ Πn then φ, ψ ∈ ∆n+1, φ → ψ ∈ Πn T and ψ → φ ∈ Σn . T T T T – φ ∈ Σn → ¬φ ∈ Πn and φ ∈ Πn → ¬φ ∈ Σn .

(ii) Quantiﬁers (for T ⊢ Z − inf)

T T – φ ∈ Σn → ∀x φ ∈ Πn T T – φ ∈ Πn → ∃x φ ∈ Σn T T – If n> 0 then φ ∈ Σn → ∃x φ ∈ Σn T T and φ ∈ Πn → ∀x φ ∈ Πn .

(iii) Bounded quantiﬁers (for T ⊢ ZF ) T T If φ is Σn or Πn then so are ∃x ∈ y φ and ∀x ∈ y φ.

(iv) Terms (if n> 0 and T ⊢ ZF )

T T T – If φ(x) ∈ ∆n then {x : φ(x)} ∈ Πn and {x ∈ z : φ(x)} ∈ ∆n . T T – If t ∈ Σn and T ⊢ ∃x x = t then t ∈ ∆n . T – Suppose T ⊢ ∃x x = t. Then if φ(y), s(y) and t are ∆n then so are φ(t), s(t), ∃x ∈ t φ, ∀x ∈ t φ and {x ∈ t : φ}.

Now that we are equipped with theorem 1.10 and lemma 1.11 we are ready to ZF give a list of terms and formulas in set theory which are ∆1 and thus absolute. The notions described by those terms and formulas, such as the ordinals, tend ZF to be easier in their manipulation. In contrast, notions which are not ∆1 , such as cardinals or the power set operation, are more vague and a variety of hard (many times unsolvable in ZFC) problems arise.

Lemma 1.12. 1.3. ABSOLUTENESS 9

ZF (i) The following terms and formulas are ∆0 thus absolute: (1) x ∈ y (9) x \ y (17) dom(R) (25) succord(x) (2) x = y (10) s(x) (18) rang(R) (26) finord(x) (3) x ⊂ y (11) trans(x) (19) funct(f) (27) ω (4) {x, y} (12) ∪ x (20) f(x) (28) n, n ∈ ω (5) hx, yi (13) ∩ x (21) f ↾ x (29) An (6) ∅ (14) ordpair(z) (22) f is 1 − 1 (30) A<ω (7) x ∪ y (15) A × B (23) ordinal(x) (8) x ∩ y (16) rel(R) (24) limord(x)

ZF (ii) If G(x, z, y¯) is a Σn term, and F (x, y¯) is the term obtained by transﬁnite ZF ∈ or < induction then, F (x, y¯) is ∆n . ZF (iii) The following terms and formulas are ∆1 thus absolute: β (1) α − 1 (4) α (7) Vω (2) α + β (5) rank(x) (3) α · β (6) T C(x)

ZF (iv) The following terms and formulas are Σ1 thus absolute for extensions: (1) x =c y (2) x ≤c y (3) cf(α)

ZF (v) The following terms and formulas are Π1 thus absolute for restrictions: (1) card(α) (4) inaccessible(α) (7) Vα (2) regular(α) (5) < wellorders x (3) limcard(α) (6) P (x)

Applications This section is quite technical but the reader should reach a level of familiarity with it. Absoluteness is a powerful tool when dealing with models of ZF . Imagine a set M which is a model of set theory. If we are interested in whether M φ, where φ is absolute, we only have to check if it is true in V instead of checking if (φ)M is true in V , which could be difficult. Absoluteness also comes in handy when somebody wants to show that a set M is a model of ZF C, since many of its axioms are ∆0 thus absolute. We will close this section with two applications. In the first one, we find some axioms of ZF C which are absolute ZF for all transitive models and in the second one, we prove that there exists a ∆1 formula which expresses the notion M ZF . Application 1.13. The following formulas and terms related to the corresponding axioms of ZF are absolute for every transitive model M:

A1. Extensionality A5. {x, y} A2. F oundation A7. ∪ x A4. ∅ A8. ω 10 CHAPTER 1. PRELIMINARIES

The following formulas and terms related to the corresponding axioms of ZF are preserved by restriction to M, for every transitive structure M:

A6. P (x) A10. < wellorders x Additionally, for every transitive model M, P M (x)= P (x) ∩ M.

The above application is useful when we have to show that a given transitive structure satisfies A4, A5, A7 and A8. It is transitive thus it satisfies A1,A2 and in order to show that it satisfies A4, A5, A7, A8, we only have to check that it contains ∅ and it is closed under the respective operations. A6 and A10 may need some more work, but at least this theorem covers the one direction of the proof. For the next application we need some definitions first. In order to express the notion of within the language of set theory we must initially do this for its formulas. This is the idea of Gödel numbering and it can be implemented in many ways.

Deﬁnition 1.14. Each formula is represented by an element of Vω using the function pq which is deﬁned by the following recursion:

(i) For atomic formulas:

– pxi = xj q = h0,i,ji

– pxi ∈ xj q = h1,i,ji

(ii) For non-atomic:

– pφ ∨ ψq = h2, pφq, pψqi – p¬φq = h3, pφqi

– p∃xi φq = h4, i, pφqi

The subset of Vω, which consists exactly of its elements that express a formula, will be denoted Form.

Deﬁnition 1.15. We give the following deﬁnitions:

(1) fm(u,f,n): “u = pφq where φ is the sentence whose structure is described by the function f in n steps”.

(2) fmla(u): “u is the G¨odel set of a formula”.

(3) s(m,g,r,f,M): “when f, g are functions, f describes the construction of a sentence of rank r and f(k) = pφmq then, g(m) contains all the r- r tuples, a, of M for which φm[a] takes its truth value according to Tarski’s deﬁnition of truth for M”.

(4) sat(u, M, ¯b): “u is the G¨odel set of a formula φ for which M φ[¯b]”. 1.3. ABSOLUTENESS 11

(5) axZFC (u): “u is a formula and it is one of the axioms of ZF C”. (6) M ZF :“M is a model of ZF ”. ZF Application 1.16. The above notions are all ∆1 thus absolute for all models of ZF . In particular M ZF is absolute for models of ZF . Proof. (1) Writing down the deﬁnition of fm we get the following formula: fm(u,f,n) ≡ funct(f) ∧ finord(n) ∧ dom(f)= n +1 ∧ f(n)= u∧ ∧ ∀m

ZF which by lemma 2 can be easily proved to be ∆1 . (2) fmla(u) is equivalent to the formula

∃n<ω ∃f ∈ Vω fm(u,f,n)

ZF which is ∆1 by lemma 2 and (1). (3) s(m,g,r,f,M) is equivalent to the following formula (this is a coding of Tarski’s truth deﬁnition) ∃i,j <ω [(f(m)= h0,i,ji ∧ g(m)= {a ∈ M r : a(i)= a(j)})∨ ∨(f(m)= h1,i,ji ∧ g(m)= {a ∈ M r : a(i) ∈ a(j)})]∨ ∨∃k,l

ZF which is ∆1 by lemma 2 and the fact that α(i|x) is an abbreviation for (a \ {hi,a(i)i}) ∪ {hi, xi}.

(4) sat(u,M,b) is equivalent to the formula

∃f,n,r ∈ Vω [fm(u,f,n) ∧ r = rank(u)∧ ∧ ∃g (funct(g) ∧ dom(g)= n +1 ∧ ∀m

ZF ZF so it is ∆1 . Thus sat(u,M,b) is also ∆1 . 12 CHAPTER 1. PRELIMINARIES

(5) axZF (u) is equivalent to the formula

fmla(u) ∧ [u = pA1q ∨ ... ∨ pA8q ∨ ∃v ∈ Vω (fmla(v) ∧ u = pvq)]

ZF which is ∆1 . What needs explanation here is pA9q(v) which is the set representing A9φ, where φ is the formula represented by v. To prove that ZF pAiq can be written as a ∆1 sentence is immediate, as the sentence just describes the procedure by which we construct the set representing axiom i. For example for A4 ≡ ∃x0 ¬∃x1 (x1 ∈ x0) pA4q = h4, 0, h3, h4, 1, h1, 1, 0iiii In the case of the axiom of replacement we must additionally replace φ with v wherever occuring in the axiom. (6) M ZF is equivalent to the formula

ZF rank(u) ∀u ∈ Vω [ax (u) → ∀b ∈ M sat(u,M,b)]

ZF which is ∆1 .

It is easy to see that the above procedure can be done for every set of ZF axioms of set theory which is deﬁned by a ∆1 formula. The reader who has some knowledge of computability theory will realize that every set of axioms S is of this kind, if and only if there exists an algorithm which decides whether a given element of Form belongs to S. A special case is when we have a ﬁnite extension of ZF .

1.4 Vα and Lα In this section we study models of set theory created by considering the sets Vα,Lα for diﬀerent kinds of α, and their unions which are classes. Some of those are models of ZF or ZF C and some others satisfy part of it but disagree in one of its axioms. This way we will get some results on the independence of the axioms of ZF and ZF C. For more on this subject see [21]. Our intention in this chapter is to give some more information on the structure of a universe of sets satisfying ZF C or part of it. We also want to raise some questions referring to models of ZF C and which will be correlated with the existence of speciﬁc large cardinals in later chapters.

Vα

We have already introduced Vα, α ∈ On as the sets forming von Neumann’s hierarchy. In the case α is a successor cardinal, they don’t present much interest as models because they fail to be closed under most of the operations appearing in ZF . On the other hand, if α is a limit ordinal Vα is nearly a model of ZF C. The following lemma provides some information on this case. 1.4. Vα AND Lα 13

Lemma 1.17.

(i) Vω ZF (C) − inf + ¬inf.

(ii) If α>ω and limord(α) then Vα ZC.

(iii) Vω+ω ZC + ¬F .

Finally, we have to notice that R = ∪α∈OnVα is a model of ZF C, something we already knew since R = V .

Lα

The sets Lα have been deﬁned in section 1.2. They are similar to the sets Vα but the notion of deﬁnability provides us with a much clearer view of the universe L. This is also the reason why many more axioms can be proved to be valid in L, such as AC and GCH, by accepting only ZF . Theorem 1.18.

(i) Lω ZF C − inf + ¬inf.

(ii) Lα ZC, if α is a limit ordinal. (iii) L ZF (iv) L V = L Proof.

(i)-(iii) The proofs are like the ones for Vα because all the notions appearing in the axioms of ZF C are deﬁnable. (iv) The formula y = def(M) is equivalent to

∀x ∈ M ∀u ∈ Vω (x ∈ y ↔ fmla(u) ∧ M u[x])

ZF ZF which is ∆1 so def(M) is a ∆1 term.

The sets La are deﬁned by transitive < induction using the term ∅ z =0 G(f,z)=  def(f(z − 1)) succord(z)  ∪α

L L V = L ↔ ∀x ∈ L ∃α ∈ L (x ∈ Lα) ↔ ∀x ∈ L ∃α ∈ L x ∈ Lα

which is true. 14 CHAPTER 1. PRELIMINARIES

V = L is a very crucial property of L. Along with the fact that Lα is absolute for transitive models of ZF , it will assist us in the proof of the next theorems.

Deﬁnition 1.19. An inner model of ZF is a transitive class (M, ∈), such that M ZF and On ⊂ M.

Theorem 1.20. L is the smallest inner model of ZF .

Proof.

• We have proved that L ZF and that ∀α ∈ On α ∈ Lα so L is an inner model of ZF .

• Since M is a model of ZF , M ∀α ∃x x = Lα, but Lα is absolute and α ∈ M for every ordinal, so ∀α Lα ∈ M → L ⊂ M.

Lemma 1.21. (G¨odel’s condensation lemma) If M ≺ Lα, limord(α), then the Mostowski collapse of M is Lβ for some β ≤ α.

Proof. The main idea is to prove ﬁrst, that there exists a sentence φL such that for every transitive model N satisfying enough of the axioms of ZF

N φL ↔ ∃α (limord(α) ∧ N = Lα).

After a step by step examination of the proof of the absoluteness of Lα we can ZF T see Lα is not only ∆1 but ∆1 where T is an appropreate, ﬁnite part of ZF . Thus if N is a transitive model satisfying T then

N ∀x ∃α x ∈ Lα ↔ ∀x ∈ N ∃α ∈ N x ∈ Lα ↔ ∃α (limord(α) ∧ N = Lα).

If M ≺ Lω then M = Lω. Else, if α>ω we must notice that the axiom of replacement was only needed in order to deﬁne some sets by ω-transitive recursion and thus show that Lα is absolute. For those cases needed it is also true in Lα. Thus since Lα Z we have that Lα T so Lα φL. If M ≺ Lα then its transitive collapse, N, N φL → ∃β N = Lβ. Since M ⊂ Lα the ordinals of M, and thus of N, belong to Lα, hence β ≤ α.

Theorem 1.22. L AC

Proof. In order to prove that L AC we only have to show, in ZF , that every set of L can be well-ordered, because then from theorem 1.18, (iv)

L (ZF + V = L) → L ∀x ∃R (R well − orders x) → L AC.

The proof is done using transitive < induction.

• L0 = ∅ is well-ordered by the empty relation. 1.4. Vα AND Lα 15

n • Let <α be a well-ordering of Lα, <α the lexicographical order, derived from <α, of ordered n-tuples,

n x<α+1 y ↔ φx

• If α is a limit ordinal and rankL(x) is the rank of x in the Lα hierarchy then we can deﬁne the following well-ordering of Lα:

x<α y ↔ rankL(x) < rankL(y)∨(rankL(x)= rankL(y)∧x

A well-ordering

Theorem 1.23. L GCH

3 Proof. We will show that if x ∈ P (ℵα), then there is a β < ℵα+1 such that x ∈ Lβ. So P (ℵα) ⊂ Lℵα+1 thus, since |Lℵα+1 | = ℵα+1,

|P (ℵα)|≤ℵα+1 → |P (ℵα)| = ℵα+1.

Provided that the above holds for every ordinal α, GCH will be true. Let x ∈ P (ℵα) and x ∈ Lγ. By the L¨ovenheim-Skolem theorem, there is a model M ≺ Lγ which satisﬁes ℵα ∪{x} ⊂ M and |M| = ℵα. By lemma 1.21, there exists a limit ordinal β ≤ γ such that the Mostowski collapse N of M is Lβ. We also have that

|β| = |Lβ| = |N| = |M| = ℵα < ℵα+1.

Finally, ℵα ⊂ M, so if F is Mostowski’s collapsing function of M, F ↾ ℵα is the identity function. But x ⊂ℵα thus x = F (x) ∈ N = Lβ.

The conception of the universe L of the constructible sets and the above theorems are due to G¨odel. They imply that ZF + AC and ZF + AC + GCH are consistent provided that ZF is. Exploring L, one can prove the relative consistency of many other principles, such as ♦ and ♦+.

3 L L We note that in this proof we are working inside L. Thus P (x) is (P (x)) , ℵα is (ℵα) etc. 16 CHAPTER 1. PRELIMINARIES

Constructibility relative to a set A Finally, we briefly state some facts about constructibility relative to a set A. This kind of consructibility can lead to two inner models, L[A] and L(A). The model L[A] is constructed as L, with the only difference that we allow the set A to participate in the procedure of defining new sets. Thus, we alter the notion of definable sets in the following way:

Deﬁnition 1.24.

defA(M)= {X ⊂ M : X is definable over (M, ∈, A ∩ M)}.

Deﬁnition 1.25. The class L[A] of the sets constructible from A is equal to

α∈On Lα[A], where S (i) L0[A]= ∅;

(ii) Lα+1[A]= defA(Lα[A]);

(iii) Lα[A]= β<α Lβ[A] if limord(α). S The following theorem contains some results relative to those we have proved for L.

Theorem 1.26. For every set A,

(i) L[A] ZF C;

(ii) L[A] ∃X V = L[X];

(iii) If M is an inner model and A ∩ M ∈ M, then L[A] ⊂ M;

(iv) If limord(α) and M ≺ (Lα[A], ∈, A ∩ Lα[A]), then the Mostowski collapse of M is Lβ for some β ≤ α;

ℵβ (v) ∃α ∀β ≥ α L[A] 2 = ℵβ+1;

(vi) If A ⊂ℵ1, then L[A] GCH.

The model L(A) is diﬀerent from L[A], as it contains A. Actually, it is the smallest inner model which contains A.

Deﬁnition 1.27.

(i) L0(A)= T C(A);

(ii) Lα+1(A)= def(Lα(A));

(iii) Lα(A)= β<α Lβ(A) if limord(α). S L(A)= α∈On Lα(A). S 1.5. MODEL THEORY 17 1.5 Model theory

In this section we sketch some model theoretical concepts which will become useful later on. The reader can refer to [2] and [21] for more details and some of the proofs which have been omitted.

Skolem functions We have already met the notion of elementary substructures while introducing absoluteness. It is quite clear that all the logical and nonlogical symbols, except from the quantiﬁers, are interpreted the same way in both structures M ⊂ N. Hence, in order to check that M ≺ N, we must examine whether the quantiﬁers are interpreted the same way between M,N. Since ∀ is equivalent to ¬∃¬, we only have to work with ∃. Thus the following holds:

Lemma 1.28. M ≺ N iﬀ for every φ ∈ Form and every n-tuple ¯b ∈ M:

∃a ∈ NN φ[a, ¯b] → ∃a ∈ M N φ[a, ¯b].

The lemma above can be used in constructing elementary substructures of a given structure M. Speciﬁcally, we can deﬁne for every formula φ(v,u1,...,un) n a function hφ : M → M such that for all a1,...,an ∈ M

M φ[hφ(a1,...,an),a1,...,an] if ∃a ∈ MM φ[a,a1,...,an];

hφ(a1,...,an)= a0 ∈ M else.

The functions satisfying those properties are called Skolem functions. The closure H(X) of a set X ⊂ M under those functions is called the Skolem hull of X and by lemma 1.28, H(X) ≺ M. Many times Skolem functions can be defined inside a structure but, even if not, we can always prove the existence of external Skolem functions for every structure M, using AC. This fact leads to the following theorem: Theorem 1.29. (Löwenhein-Skolem) Suppose L is a language and M a structure for L. Then, (i) (downward) for every cardinal |L| ≤ κ ≤ |M|, there is a stucture N ≺ M, such that |N| = κ; (ii) (upward) for every cardinal κ ≥ |M|, there is a structure N ≻ M, such that |N| = κ. We define now the notion of elementary embeddings, which will play a crucial role in our study. Definition 1.30. An elementary embedding of M into N, is a function f : M → N such that: (i) f is an isomorphism between M and f ′′M; 18 CHAPTER 1. PRELIMINARIES

(ii) M ≺ N.

We will use the notation f : M ≺ N to express the fact that f is an elementary embedding of M into N.

Elementary chains

Deﬁnition 1.31. Every sequence of structures {Mβ}β<α, such that for all

β1 <β2 < α, Mβ1 ≺ Mβ2 , is called an elementary chain.

For every elementary chain we may deﬁne its union M = β<α Mβ: S (i) M = β<α Mβ; S M Mβ (ii) P = β<α P ; S M Mβ (iii) F = β<α F ; S (iv) cM = cMβ for all β < α.

It is straightforward to check that M is well deﬁned and that it is a structure. What is also true is the following:

Theorem 1.32. (Elementary chain theorem) For every chain {Mβ}β<α and every γ < α, Mγ ≺ β<α Mβ. S Ultraproducts A very useful method in constructing structures is the one using ultraproducts. In order to understand it we must first define filters and ultrafilters. A filter on a set S is a subset of P (S), which contains those subsets of S that are in some sense “large” and can be considered to be nearly the same as S. The aspect of what is large, depends on the nature of each specific filter.

Deﬁnition 1.33. Let S be a set and F ⊂ P (S). F is a ﬁlter on S if:

(i) ∅ ∈/ F and S ∈ F ;

(ii) X, Y ∈ S → X ∩ Y ∈ S;

(iii) X ∈ S ∧ X ⊂ Y ⊂ S → Y ∈ S.

Some examples of filters are the Fréchet filter on N, which contains the subsets of N that have finite complements, and the filter on R, which contains all the subsets of the reals having Lebesque measure 1. It is up to the reader to establish why those two are filters and understand the meaning of “large” in both cases.

Definition 1.34. A filter U on S is called an ultrafilter, if for every X ⊂ S X ∈ U or S \ X ∈ U. 1.5. MODEL THEORY 19

Our intention is to use the notion of ultrafilter to define some sort of product of structures, the same way we construct products of groups or topological spaces. Suppose {Mx : x ∈ S} is a family of structures and U an ultrafilter. We define the equivalence relation =U on x∈S Mx so that, f =U g ↔{x ∈ S : f(x) = g(x)} ∈ U. The ultraproduct of theQ structures Mx is then defined the following way:

Deﬁnition 1.35. For every family of structures {Mx : x ∈ S} and every ultra- ﬁlter U on S, M = U Mx is the structure with the following properties: Q (i) M = x∈S Mx/ =U ;

M Q Mx (ii) P ([f1],..., [fn]) ↔{x ∈ S : P (f1(x),...,fn(x)} ∈ U for every predi- cate symbol P ;

M Mx (iii) F ([f1],..., [fn]) = [f] ↔ {x ∈ S : F (f1(x),...,fn(x)) = f(x)} ∈ U, for every function symbol F ;

(iv) cM = [f], ∀x ∈ S f(x)= cMx , for all constant symbols c.

It is not diﬃcult to check that the relation =U is an equivalence relation and that M is indeed a structure, as those concepts depend solely on the properties of ultraﬁlters. Again by applying those properties, we can get the following theorem, which gives us a way of calculating M from Mx , x ∈ S.

Theorem 1.36. (Lo´s) Let M = U Mx be as above. Then, for every formula φ and [f1],..., [fn] ∈ M, Q

M φ[[f1],..., [fn]] ↔{x ∈ S : Mx φ[f1(x),...,fn(x)]} ∈ U.

If for every x ∈ SMx = N, where N is a speciﬁc structure, we call the ultraproduct U Mx an ultrapower and denote it by Ult(N,U). It is straightforward by theQ theorem above that N ≡ Ult(N,U), i.e. they satisfy the same sentences. Moreover we can deﬁne the canonical embedding j : N → Ult(N,U), j(x) = [cx], where cx is a constant function such that ∀y cx(y)= x. This -again by theorem 1.36- is an elementary embedding.

Reflection principles A reflection principle, in general, asserts that a certain property of the universe V is absolute for V , M, where M is a model of ZF C. The property can be expressed by a first order formula, a higher order formula, a type, or anything else we could imagine. Most of the times the validity of reflection principles, for some specific kind of properties, cannot be proved from ZF C, hence they could n be proposed as new axioms. This is the case of Πm-indiscernibles, which we will study in the next chapter. For the time being, we state the following theorem, that contains three similar reflection principles, provable from ZF or ZF C.

Theorem 1.37. For every formula φ and every set X: 20 CHAPTER 1. PRELIMINARIES

(i) There is an ordinal α, such that X ⊂ Vα and φ is absolute for V , Vα.

(ii) There is a structure M, such that X ⊂ M, |M| ≤ |X|·ℵ0 and φ is absolute for V , M.

(iii) There is a transitive structure M such that |M|≤ℵ0 and φ is absolute for V , M. Proof. (i) We deﬁne a function for φ using Scott’s trick, which looks like a Skolem function, and by repeatingly applying it to sets of von Newmann’s hierarchy we get Vα. (ii) It’s the same as before, though we use Skolem’s functions this time, which exist by AC. (iii) We apply Mostowski’s collapsing lemma to (ii) and get the needed structure. For more details, see [11] and [8]. Chapter 2

Some smaller large cardinals

As we have seen, V can be approached by von Newmann’s cumulative hierarchy, thus we could say that it is fully described by the class On and the powerset op- eration1. This way, we can imagine that the size of On determines the “length” of V and the size of P (for several x) determines its “width”. In this chapter we are going to see some examples of large cardinals. Although there is not a certain definition for this concept, we could say that large cardinals are cardinals whose existence cannot be proved from ZF C and they extend the universe V in length. Throughout the text, we will notice that the categories of large cardinals we are going to introduce, appear in a sequence of increasing strength (a large cardinal property a is stronger than b if ZF C + ∃κ a(κ) ⊢ ∃κ b(κ), e.g. ZF C + ∃κ Mahlo(κ) ⊢ ∃κ inaccessible(κ)). We begin by citing some aspects of cardinal arithmetic, necessary for the definitions we are going to give and for some of the proofs. We define then inaccessible cardinals, which cannot be defined using the operations described in the first chapter, thus in some way they cannot be approached by the means of ZF C. We will see that their existence is connected to the existence of models of ZF C inside V . By requiring the existence of more and more inaccessible cardinals, we will come up with Mahlo cardinals which demand the existence of a stationary set of inaccessible cardinals. From this point on, we will need new methods n for finding even stronger categories of large cardinals. The Πm-inaccessibility 1 will provide us with the weakly compact cardinals (Π1-inaccessible), along with some useful characterizations or properties of other categories of large cardinals. Weakly compact cardinals, owe their name to their connection with the

1Those two notions are not fully determined by ZF C, therefore V could vary in length and width. Using forcing, we can show that the cardinality of P (X) may take various diﬀerent values greater or equal to |X|. Hence, we can not even count the increase in size the powerset operation imposes. As for On, the more large cardinals we require to exist, the larger On gets. The only diﬀerence here from the case of forcing, is that we can not prove in ZF C that the existence of large cardinals is consistent with ZF C.

21 22 CHAPTER 2. SOME SMALLER LARGE CARDINALS

property of weak compactness of an infinitary logic Lκ,ω, where κ is an infinite cardinal. Finally we introduce Erd˝os and Ramsey cardinals, which are defined using combinatorial partition properties. We call the cardinals defined in this section (except from Ramsey cardinals) “small”, since their existence is relatively consistent with the assumption V = L. Hence, they are not strong enough to cause drastic changes to the structure of V . In the contrary, the existence of Ramsey cardinals or of the large cardinals introduced in the next chapters, will have V 6= L as a consequence. Even more, if one of them exists, V will be much more complicated than L. For more on this check chapter 4.

2.1 Cardinal arithmetic

This section contains some basic knowledge on cardinal arithmetic. All the material covered here can be found in any introductive book on set theory.

+, ·, and +, ·, Pand areQ some of the functions defined on cardinals that will be used manyP timesQ in our study. Below we give their definitions, ways of calculating them and some of their properties. Definition 2.1.

(i) The sum of two cardinals κ and λ, κ + λ, is deﬁned by:

κ + λ = |κ ×{0} ∪ λ ×{1}|.

(ii) The product of two cardinals κ and λ, κ · λ, is deﬁned by:

κ · λ = |κ × λ|.

Theorem 2.2. The class On×On can be well-ordered by the following relation:

hα1, α2i ⊳ hβ1,β2i ↔ max{α1, α2} < max{β1,β2}∨

∨[max{α1, α2} = max{β1,β2}∧

∧(α1 < α2 ∨ (α1 = α2 ∧ β1 <β2))].

Moreover, for every cardinal κ, hκ × κ,⊳i =o κ. Proof. ⊳ is a well-ordering of On × On. If A is a subclass of it, then the set

{α ∈ On : ∃ hα1, α2i ∈ On × On min{α1, α2} = α} has a minimum element β. The minimum of the set

{hα1, α2i ∈ On × On : min{α1, α2} = β} 2.1. CARDINAL ARITHMETIC 23 ordered by the lexicographical ordering, is a minimum for A. The reduction of ⊳ to κ × κ is a well-ordering for κ × κ and κ ≤o hκ × κ,⊳i. Thus we have to show that hκ × κ,⊳i≤o κ. If this is not true and κ is the least cardinal with the property κ

∃α, β ∈ κ κ =o seg⊳(α, β) and if γ = max{α, β}

seg⊳(α, β) ≤o hγ × γ,⊳i =o hγ,

This means that κ<κ which is a contradiction. Corollary 2.3. For all cardinals κ, λ and κ + λ = κ · λ = max{κ, λ}. Proof. max{κ, λ}≤ κ + λ ≤ κ · λ so we only need to prove that κ · λ ≤ max{κ, λ}, but this is true since κ · λ ≤ max{κ, λ}· max{κ, λ}≤o max{κ, λ}. The above corollary reveals that the computation of κ+λ and κ×λ is trivial. Deﬁnition 2.4.

(i) The sum of the cardinals belonging to the set {κi : i ∈ I} is deﬁned as:

κi = | ∪i∈I (κi ×{i})|. Xi∈I

(ii) The product of the cardinals belonging to the set {κi : i ∈ I} is defined as: κi = | κi| Yi∈I Yi∈I (the second product is the set product and it is different from the first one) We have to note here that the definition of requires in most cases the axiom of choice in order to be used. This is becauseQ every sequence {αi}i∈I , ai ∈ κi is a choice function on the collection of the cardinals κi. From now on we will take AC as for granted and we will just keep in mind that it is essential in most of the results that follow. Some basic properties of and are stated below: P Q Lemma 2.5.

(i) α<λ κ = λ · κ;

P λ (ii) α<λ κ = κ ; Q (iii) κ = ( κ ), where I is the disjoint union ∪ A . i∈I i j∈J i∈Aj i j∈J j P P P 24 CHAPTER 2. SOME SMALLER LARGE CARDINALS

λ λ (iv) α<µ κα = ( α<µ κα) ); Q Q λα Pα<µ λα (v) α<µ κ = κ ; Q (vi) κ = ( κ ), where I is the disjoint union ∪ A . i∈I i j∈J i∈Aj i j∈J j Q Q Q The computation of i∈I κi is also quite trivial due to the following theorem: P Theorem 2.6. If ω < λ and ∀α < λ 0 <κα then

κα = λ · sup κα. α<λX α<λ

The computation of is not at all trivial, because it is related to the exponentiation function. TheQ next theorem reveals this connection.

Theorem 2.7. If λ is an inﬁnite cardinal and hκα : α < λi is nondecreasing sequence of nonzero cardinals, then

λ κα = (sup κα) α<λ α<λY

Since is related to exponentiation and to multiplication, which is much weaker, itQ is natural to search for a result generalizingP Cantor’s theorem ℵα < 2ℵα . K¨onig’s theorem is such a result.

Theorem 2.8. (K¨onig) Suppose that ∀α<µκα < λα. Then

κα < λα. α<µX α<µY

Cofinality A useful notion in the study of cardinals is cofinality. The cofinality of a cardinal is the smallest cardinal which can be used to approach it. Below we formally define it and state some of its properties.

Deﬁnition 2.9. Suppose κ is a cardinal and α ≤ κ is an ordinal.

(i) A function f : α → κ is coﬁnal, if sup(f ′′α)= κ.

(ii) The coﬁnality of κ, is the least ordinal α, such that there is a coﬁnal function f : α → κ. We denote it by cf(κ).

Another characterization of coﬁnality is the following:

Lemma 2.10. The coﬁnality of a cardinal κ is the least ordinal α, such that there is a family of cardinals {κβ : β < α}, ∀β<ακβ <κ, having the property β<α κβ = κ. P 2.1. CARDINAL ARITHMETIC 25

It is easy to check that the cofinality of a cardinal is a cardinal and that cf(cf(κ)) = cf(κ). The following lemma is useful in calculating the cofinality of a cardinal. Lemma 2.11. Suppose that κ, λ are cardinals. If there is a nondecreasing cofinal function f : κ → λ, then cf(κ)= cf(λ).

As an example of the above, we see that cf(ℵ0)= ℵ0 since it must be inﬁnite.

Thus, taking under consideration that f : ℵ0 → ℵω0 , f(n) = ℵn is increasing and cofinal, cf(ℵω0 )= cf(ℵ0)= ℵ0. We end our discussion on cofinality, by stating the following result which is a corollary of König’s theorem. Corollary 2.12. For all cardinals κ, λ: (i) cf(κλ) > λ; (ii) κcf(λ) >κ.

[A]λ, [A]<λ and SCH We finally define the two following two notions and state the singular cardinal hypothesis (SCH). Definition 2.13.

(i) [A]κ = {X ⊂ A : |X| = κ}. (ii) [A]<κ = {X ⊂ A : |X| <κ}.

<κ Many times we denote [A] by Pκ(A). Lemma 2.14. |A|≥ λ → |[A]κ| = |A|κ. Deﬁnition 2.15. SGH : For every singular cardinal κ

2cf(κ) <κ → κcf(κ) = κ+.

The SCH is a weaker version of GCH and by the following theorem it is clear that it simpliﬁes the calculation of the exponential function. Theorem 2.16. Let SCH hold. (i) If κ is singular, then – 2κ =2<κ, if 2<κ =2λ, λ<κ; – 2κ = (2<κ)+, else. (ii) If κ, λ ≥ ω, then – κλ =2λ, if κ ≤ 2λ; 26 CHAPTER 2. SOME SMALLER LARGE CARDINALS

– κλ = κ, if 2λ <κ and λ

2.2 Inaccessible cardinals

Definition 2.18. (i) A cardinal κ is called regular, if cf(κ)= κ. (ii) A cardinal κ is strong limit, if for every λ<κ, 2λ <κ. Definition 2.19. The cardinal κ is called inaccessible if it is strong limit and regular. By its definition, if there exists an inaccessible cardinal κ, then it ought to be quite a big cardinal. One easy way to see this, is by noticing that iκ = κ and that the smallest cardinal having this property is the union of the following sequence:

α0 =i0 i αn+1 = αn

Inaccessible cardinals and models of ZFC The main property that inaccessible cardinals satisfy, is that they deﬁne models of ZF C in the cumulative hierarchy Vα.

Theorem 2.20. If κ is an inaccessible cardinal, then Vκ ZF C. Proof. Since κ is a cardinal, it is also a limit ordinal. We know from chapter 1 that in this case Vκ ZC, thus we only need to show that Vκ F . Let us assume that x is an arbitrary set and that φ is a formula which deﬁnes Vκ′′ a function. Every element of Fφ x belongs to Vα for some α<κ so

Vκ′′ Fφ x ⊂ Vβ

Vκ′′ where β = sup{α : ∃y ∈ Fφ x y ∈ Vα}. κ is a strong limit cardinal thus we can show using transﬁnite induction that Vκ′′ |x| <κ for every x ∈ Vκ. As a result of that |Fφ x| < κ and provided that κ Vκ′′ is regular, β<κ which means that Fφ x ∈ Vκ. 2.2. INACCESSIBLE CARDINALS 27

The inverse of this theorem is not true. This can be justiﬁed by considering the case where there exists at least one inaccessible cardinal. If κ is the least inaccessible cardinal, then we can construct the following elementary chain {An}n<ω of models:

A0 =Vω

An+1 =Vβ, β = inf{α<κ : H(An) ⊂ Vα}

where H(x) is the Skolem hull of x in the model Vκ. By induction, An ⊂ Vαn , αn <κ, so the sequence is well deﬁned and additionally ∪n<ωAn = Vα, α<κ. By the elementary chain theorem, Vα ≺ Vκ. This means that Vα ZF C and α is not inaccessible. Despite of that, if we formulate the axioms of ZF C in second order logic, so that the axiom of replacement becomes

∀F ∀x ∃y y = F ′′x then the inverse of 2.20 is also true.

2 Theorem 2.21. The cardinal κ is inaccessible iﬀ Vκ ZF C . Proof.

→ This can be done as in the proof of theorem 1.4, since we only used there Vκ the fact that Fφ is a function.

2 ← Suppose Vκ ZF C . κ is regular because if it was not, then there would exist a cardinal λ<κ and an increasing function f : λ 7→ κ such that ∪f ′′λ = κ. But then, by second-order replacement

′′ ∪f λ ∈ Vκ → κ ∈ Vκ

which is a contradiction. κ is also strong limit because if not, then there would be a cardinal λ<κ such that κ ≤ 2λ. This means that there is function f : P (λ) → κ onto κ. We already know that P (λ) ∈ Vκ so, by second-order replacement, ′′ κ = f P (λ) ∈ Vκ which is again a contradiction.

This connection of inaccessible cardinals with the existence of models of ZF C2 in the cumulative hierarchy, provides us with a way of understanding those cardinals. Suppose that we have a universe V , where none of the sets 2 Vα are models of ZF C . This is the smallest, in length, closure of the empty set, under the operations described in 1.1 for ZF C2. If we rename V so that V = Vκ (κ∈ / V ) and consider it to be a set, we can produce the smallest in length closure V ′ of V ∪{V } under the operations described above. V ′ will be a new universe where there will exist exactly one inaccessible cardinal. Thus the 28 CHAPTER 2. SOME SMALLER LARGE CARDINALS existence of inaccessible cardinals is a way to express the number of times we have iterated this closure to the empty set. An analogue of this situation is in the study of the models of Peano arithmetic, where the standard model is the smallest one in length, but we can add more elements, using Skolem functions, and produce nonstandard models which are the one extension of the other. Of course, the diﬀerence here is that the existence of those models is granted by ZF C but in the universe V the existence of the closures is based just on intuition. As it is expected, the existence of inaccessible cardinals is not provable in ZF C. Even more, the relative consistency of their existence with ZF C can not be proved in ZF C. We will denote by IC the statement that claims the existence of an inaccessible cardinal. Theorem 2.22.

(i) ZF C 6⊢ IC. (ii) ZF C 6⊢ [cons(ZF C) → cons(ZF C + IC)]. Proof. (i) If the opposite was true, then

ZF C ⊢ IC →ZF C ⊢ ∃κ (Vκ ZF C) → →ZF C ⊢ cons(ZF C)

which is impossible according to G¨odel’s second incompleteness theorem. (ii) If ZF C is consistent, and we could prove that

cons(ZF C) → cons(ZF C + IC)

then ZF C + IC would also be consistent. We would have

ZF C + IC ⊢ [cons(ZF C) → cons(ZF C + IC)]

and since ZF C +IC ⊢ cons(ZF C) we would be able to prove that ZF C + IC ⊢ cons(ZF C+IC) which is again a contradiction due to G¨odel’s second incompleteness theorem.

1 Inaccessible cardinals as Σ1-indescribables Finally, another characterization of inaccessible cardinals is the following:

1 Theorem 2.23. κ is inaccessible iﬀ it is Σ1-indescribable. Before proving this theorem, we state a characterization of inaccessible cardinals by elementary embeddings. 2.2. INACCESSIBLE CARDINALS 29

Lemma 2.24. κ is inaccessible iﬀ for any R ⊂ Vκ there is an α such that

(Vα, ∈, R ∩ Vα) ≺ (Vκ, ∈, R).

Proof.

→ We deﬁne Skolem functions on the model (Vκ, ∈, R) and form the following elementary chain:

A0 =(Vω, ∈, R ∩ Vω)

An+1 =(Vβ, ∈, R ∩ Vβ ), β = inf{α<κ : H(An) ⊂ Vα}.

Every An is a subset of Vκ, because κ is inaccessible and moreover the union of the chain is Vα for some α<κ. By the elementary chain theorem (Vα, ∈, R ∩ Vα) ≺ (Vκ, ∈, R).

← – κ is regular because in the opposite case there would exist an ordinal β and an increasing function f : β 7→ κ onto κ. If R = {β} ∪ f, then Vα ∃x ∈ R ordinal(x) which means that β ∈ Vα and this is a contradiction because (f ∩ Vα)(β)= κ ∈ Vα. – κ is also strong limit. On the contrary, there would exist a λ<κ such that κ ≤ 2λ thus also a function f : P (λ) 7→ κ onto κ. If R = {λ +1} ∪ f, then again λ +1 ∈ Vα → λ < α thus P (λ) ∈ Vα which is a contradiction.

Proof. (theorem 2.23) 1 By lemma 2.24, if κ is inaccessible then it is Π0-indescribable. The inverse is also true. In order to show this, one must go through the second part of the proof of lemma 2.24 and notice that instead of the elementary submodel, we could just use a submodel of (Vκ, ∈, R) which satisﬁes a speciﬁc formula. 1 1 Thus we only need to show that Π0-indescribability is equivalent to Σ1- indescribability:

1 1 • If a formula φ is Σ1 then it is also Π0 so the one direction is obvious.

1 • Suppose that κ is Π0-indescribable and let (Vκ, ∈, R) ∃X φ(X). Then for some S ⊂ Vκ:

(Vκ, ∈, R) φ(S) →(Vκ, ∈, hR,Si) φ(S) →

→∃α<κ (Vα, ∈, hR ∩ Vα,S ∩ Vαi) φ(S) →

→∃α<κ (Vα, ∈, R ∩ Vα) ∃X φ(X)

1 thus κ is Σ1-indescribable. 30 CHAPTER 2. SOME SMALLER LARGE CARDINALS 2.3 Mahlo cardinals

In the previous section we introduced inaccessible cardinals and described the way they extend the universe of sets. Since we can extend the universe by adding one inaccessible cardinal to it, we could do the same by adding another one and repeat this procedure many times. Up to this point, the universe has been extended only by the use of inaccessible cardinals. Thus, in order to get one step further, we have to introduce a new kind of cardinals, such that the set of inaccessible cardinals below them is unbounded. This means that if κ is one of those cardinals, it must be an inaccessible which is a strong limit of inaccessible cardinals. Again, by repeating the new procedure we can create new cardinals which are strong limits of the ones of the previous kind. This way we get the following sequence of classes:

P0 ={κ : inaccessible(κ)}

Pα+1 ={κ : κ strong limit of Pα}

Pα = ∩β<α Pβ if limord(α)

The cardinals belonging to Pα are called α-inaccessible and πα is the least element of Pα. Someone may notice that we have not defined a set where the ordinal α belongs. This was done on purpose because while extending the universe with new cardinals, new ordinals are being created thus we can further extend this sequence. As before we may introduce a new kind of cardinals κ, such that κ ∈ Pκ, and build inaccessible cardinals over them and go on this way. In an attempt of finding an upper bound to this way of extending the universe, Mahlo cardinals were introduced. We will define those large cardinals below, but before we do that we will give some information on the closed unbounded (club) filter and on stationary sets. Definition 2.25. Let κ be a regular uncountable cardinal and X ⊂ κ. (i) X is unbounded in κ, if sup(X)= κ.

(ii) X is closed, if for every increasing sequence {xβ}β<α of elements of X, β<α xβ ∈ X. S It is not hard to prove that for any given regular uncountable cardinal, the closed and unbounded subsets of κ form a κ-complete2 filter on κ which is closed under diagonal intersections3. This filter is called the closed undounded (club) filter on κ. Let F be the club filter on κ. Since it is not an ultrafilter, F divides P (κ) into three categories; the subsets of κ that belong to F (big sets), their

2Closed under α-intersections, α<κ. 3 The diagonal intersection of a family of sets of ordinals {Xβ}β<α is deﬁned as follows:

△β<αXβ = {γ < α : γ ∈ \ Xβ } β<γ 2.3. MAHLO CARDINALS 31 complements which belong to I (small sets), and the rest of P (κ) (sets which are not small or big). The latter, together with the sets of F , are called stationary. We formally define them in the following way: Definition 2.26. Let κ be as in 2.25. A set S is a stationary subset of κ, if for every X belonging to the club filter of κ, S ∩ X 6= ∅. There are two important theorems on stationary sets which should be stated, in order to present the richness of their structure. Definition 2.27. A function f : X → X, where X is a set of ordinals, is regressive if for every α ∈ X f(α) < α. S Theorem 2.28. (Fodor) Suppose f is a regressive function on a stationary subset S of κ. Then, there is a stationary set T ⊂ S, such that f is constant on T . Theorem 2.29. (Solovay) Every stationary subset of κ, is equal to the union of κ disjoint stationary subsets of κ. Definition 2.30. κ is a Mahlo cardinal if the set {λ<κ : inaccessible(λ)} is stationary in κ. This is a quite reasonable definition for this notion. One could require that the set of inaccessible cardinals in κ is club, in order to ensure that they are unbounded in κ and on the same time, that if we consider one of the sequences mentioned before, its limit is inside κ. Unfortunately, this way we would have inaccessible cardinals with cofinality ℵ0, which is a contradiction, so instead we require that the set of inaccessible cardinals in κ intersects all the club sets of κ. This way the notion of stationary sets is introduced and it is sufficient to ensure the preceding facts. As an example we prove the following: Theorem 2.31. If κ is a Mahlo cardinal, then κ is κ-inaccessible.

Proof. Let S be the set κ ∩ P0 and Cα the sets deﬁned by transﬁnite recursion:

C0 =κ

Cα+1 =limitpoints(Cα ∩ S) \{κ}

Cα = ∩β<α Cα

κ is a club set. If Cα is club then Cα+1 ∩ S is stationary so the set of its limit points is club. If α<κ then, since the club ﬁlter is κ-complete, ∩β<αCα is also club. By transﬁnite induction,

• C0 ∩ S = κ ∩ P0

• Cα+1 ∩ S = limitpoints(κ ∩ Pα \{κ}) ∩ S = κ ∩ Pα+1

• Cα ∩ S = ∩β<αPβ ∩ S = ∩β<αPβ ∩ κ = κ ∩ Pα 32 CHAPTER 2. SOME SMALLER LARGE CARDINALS

Just as with inaccessible cardinals, we can express the existence of a Mahlo cardinal using elementary embeddings.

Theorem 2.32. κ is Mahlo iﬀ for every R ⊂ Vκ there is an inaccessible λ<κ such that (Vλ, ∈, R ∩ Vλ) ≺ (Vκ, ∈, R).

Proof. If not, there would be a C belonging to the club ﬁlter of κ, containing no inaccessible cardinals. By setting R = C, we get that (C ∩ λ) ∩ R = C ∩ λ is unbounded thus λ ∈ C which is not true.

Another important fact is that the existence of a Mahlo cardinal is not implied by the existence of an inaccessible cardinal.

Theorem 2.33.

(i) ZF C + IC 6⊢ ∃κ Mahlo(κ)

(ii) If there exist Mahlo cardinals, then the least of them is greater than the least inaccessible cardinal.

(iii) Every Mahlo cardinal is inaccessible.

Proof. (i) Let κ be the least Mahlo cardinal. Vκ is a model of ZF C + IC where no Mahlo cardinals exist.

(ii)-(iii) They are straightforward from the deﬁnition of Mahlo cardinals.

We could even further iterate the methods used before and create bigger cardinals but unfortunately we can not create all the known large cardinals this way. We will give a last example of this method by introducing the sequence of κ-Mahlo cardinals:

R0 ={κ : inaccessible(κ)}

Rα+1 ={κ : {λ<κ : λ is Mahlo} is stationary in R}

Rα ={κ : ∀β<ακ ∈ Rβ if limord(α)}

We denote the least element of Rα by ρα. Finally, we note that by dropping the request that the inaccessible or Mahlo cardinals are strong limits, we get their weak forms, i.e.

Deﬁnition 2.34.

(i) A cardinal is called weakly inaccessible if it is limit and regular.

(ii) A cardinal is called weakly Mahlo if the set of weakly inaccessible cardinals below it is stationary. 2.4. WEAKLY COMPACT, ERDOS˝ AND RAMSEY CARDINALS 33 2.4 Weakly compact, Erd˝os and Ramsey cardinals

In this section we are going to give a brief introduction to weakly compact, Erd˝os cardinals and Ramsey cardinals. Their common feature, is that they can be defined using the same kind of partition properties. We define below this kind of properties and provide some results describing some cases where they are true or they are not true. Definition 2.35. (i) A partition of a set X into λ pieces is a function f : X → λ.4 If f is a partition of [κ]n into m pieces, then a set Y ⊂ κ is called homogeneous if f is constant on Y .

n n (ii) κ → (α)λ is true, if every partition of [κ] into λ pieces has a homogeneous set of order type α.

<ω <ω (iii) κ → (α)λ is true if every partition of [κ] into λ pieces has a homogeneous set of order type α. In every notation above, we drop the subscript λ whenever it is equal to 2.

n Theorem 2.36. (Ramsey) For all integers m,n, ℵ0 → (ℵ0)m. Lemma 2.37. For all cardinals κ:

κ 2 (i) 2 6→ (ℵ0)κ; (ii) 2κ 6→ (κ+)2. Theorem 2.38. (Erd˝os-Rado) i+ n+1 n → (ℵ1)ω .

Weakly compact cardinals Definition 2.39. A cardinal, κ, is called weakly compact if it satisfies the 5 weak compactness theorem in the language Lκ,κ . Specifically, for every set of sentences Σ such that |Σ|≤ κ, if every S ⊂ Σ, |S| <κ has a model, then Σ has a model. We are going to give three equivalent definitions of weakly compact cardinals. The proof of their equivalence can be found in [8] and [10]. Theorem 2.40. The following are equivalent: (i) κ is weakly compact.

4This is equivalent with the usual definition of a partition, since f divides X to the λ parts f −1(0),f −1(1),.... 5For a definition and more information on infinitary languages see [8] or [1]. 34 CHAPTER 2. SOME SMALLER LARGE CARDINALS

(ii) ω<κ and κ → (κ)2.

1 (iii) κ is Π1-indescribable.

(iv) For every R ⊂ Vκ there is an transitive set X 6= Vκ and an S ⊂ X such that (Vκ, ∈, R) ≺ (X, ∈,S). Weakly compact cardinals are stronger than Mahlo cardinals in the following sense:

Theorem 2.41. The set of Mahlo cardinals below a weakly compact cardinal κ is stationary thus:

(i) ZF C + ∃κ Mahlo(κ) 6⊢ ∃λ w.compact(λ)

(ii) If there exist weakly compact cardinals, the least of them is greater than the least Mahlo cardinal.

(iii) Every weakly compact cardinal is Mahlo.

1 1 Proof. κ is Π1 thus Π0, which means that it is inaccessible. If C is in the club ﬁlter of κ, then (Vκ, ∈, C) satisﬁes the sentence

ZF C2 ∧ unbounded(C).

By the hypothesis, there is an α<κ satisfying the same sentence for C ∩ Vα. Since Vα ZF C, α is inaccessible and additionally C is unbounded in α, so C ∩ α 6= ∅. Thus, κ is Mahlo. Using the same argument and the sentence

∀X(closed(X) ∧ unbounded(X) → ∃λ ∈ X inaccessible(λ)) we can prove that the set of Mahlo cardinals below κ is stationary. (i),(ii) and (iii), are immediate consequences of the above result (also see theorem 2.33).

Erd˝os cardinals Deﬁnition 2.42.

<ω (i) Hα = {κ : κ → (α) };

(ii) The Erd˝os cardinal ηα, α ∈ On, is the least cardinal in Hα. There is an interesting connection between Erd˝os cardinals and the axiom of constructability. In particular, the existence of ηω is consistent with V = L though ηω1 is not. We shall see in the next chapter that this is related with the elementary embeddings of L. Below we state, without proof, some properties of Erd˝os cardinals. For the proof see [8] or [10]. 2.4. WEAKLY COMPACT, ERDOS˝ AND RAMSEY CARDINALS 35

Theorem 2.43.

<ω (i) ηα → (α)β for every β < ηα

(ii) ηα < ηβ for all α<β in On The next theorem settles the relation between weakly compact and Erd˝os cardinals. Theorem 2.44.

(i) If ηω exists, then there is a weakly compact cardinal below ηω. (ii) ZF C + ∃κ w.compact(κ) 6⊢ ∃λ Erd˝os(λ) (iii) Every Erd˝os cardinal is weakly compact.

We left for the end the following result which implies that ηω is consistent with V = L.

Theorem 2.45. κ ∈ Hω → L κ ∈ Hω for all cardinals κ. In particular,

∃κ κ = ηω → L ∃κ κ = ηω.

On the other hand Hω1 6= ∅ → V 6= L, thus the point where the universe no longer has chance of being constructible, is between ηω and ηω1 .

Ramsey cardinals Ramsey cardinals can be viewed as a result of diagonalizing the sequence of Erd˝os cardinals. Specifically, Definition 2.46. A cardinal κ is called a Ramsey cardinal if κ → (κ)<ω. The existence of a Ramsey cardinal κ, implies the existence of κ Erd˝os cardinals below κ. Hence, Ramsey cardinals are stronger than Erd˝os. Ramsey cardinals are the strongest ones we have encountered up to now. In order to get further up we need a new notion capable of producing even stronger large cardinals. The notion we are going to use is that of elementary embeddings of the form V → M. In the next chapter we will use it to define measurable cardinals, and in the last two chapters we will extend it as far as possible and try to reach its limitations. 36 CHAPTER 2. SOME SMALLER LARGE CARDINALS Chapter 3

Measurable cardinals and elementary embeddings of V

3.1 Aspects of measurability

This section serves as a brief reference to the foundations of the theory of measurability, introduced by Lebesque. Lebesque’s attempt to deﬁne a translation- invariant measure on R was unsuccessful and later work, by Vitali, showed that such a measure can not exist (provided that AC holds). A generalization of the notion of measure, by Banach, raised new possibilities for the existence of such a measure, which is not restricted by translation-invariance. Finally, Ulam realized that there are essentially two types of measures: the ones taking values on a dense subset of [0, 1], called atomless, and the ones whose valuation is restricted on {0, 1}, called two-valued. The existence of measurable cardinals is equivalent to the existence of two-valued measures.

The measure problem A measure, on a set S, as it was deﬁned by Banach, is a function m : P (S) → [0, 1] having the following properties:

(i) m(S) = 1;

(ii) m({x}) = 0, forall x ∈ S (nontriviality);

(iii) m( n Xn)= n m(Xn), ∀n ∈ ω Xn ⊂ S and ∀i, j ∈ ω Xi ∩ Xj = ∅. S P Such a measure is called σ-additive, because of property (iii). If the same property holds for all subsets of P (S) with cardinality less than κ, then the corresponding measure is called κ-additive.

37 38CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

We are now going to give some deﬁnitions. They are going to assist us in separating the σ-additive measures into two categories. Deﬁnition 3.1.

(i) A measure m is called two valued if m′′S = {0, 1}.

(ii) A set A ⊂ S is called an atom if m(A) > 0 and ∀B ⊂ A (m(B) = 0 ∨ m(B)= m(A)).

(iii) A measure is called atomless if it has no atoms.

A simple but useful lemma in measure theory is the following:

Lemma 3.2. Let S be a set and m a σ-additive measure on it. Every set of pairwise disjoint sets {Xα : α<γ} ⊂ P (S), such that m(Xα) > 0 for every α<γ, is countable.

1 Proof. If not, there would be an n such that the set A = {Xα : α<γ ∧ n < m(Xα)} is uncountable. By σ-additivity, the union of more than n elements of A has measure more than 1 which is a contradiction.

Lemma 3.3. If there exists a measure m on S, then S can be partitioned into, at most, a set B and ω sets An,n<ω with the following properties:

m(X) (i) the measure mB : P (B) → [0, 1], mB(X)= m(B) is atomless;

(ii) the measures m : P (A ) → [0, 1],m (X)= m(X) are two-valued. An n An m(An)

Proof. We deﬁne, by transﬁnite induction and DC, the sequence Cα ⊂ S, α< ω1,

Cα = an atom on the set Bα = S \ Cα with measure mBα ; α<ω[ 1 or ∅ if there exists none .

By lemma 3.2, only countable sets of the sequence Cαn = An, n<ω are nonempty thus B = S \ n<ω An contains no atoms of S as subsets. It is straightforward that the precedingS sets have the required properties. The above lemma suggests that we only have to deal with atomless and two- valued measures. We will restrict ourselves on measures deﬁned on a cardinal κ and investigate both cases. In order to do this though, our measure must be κ-additive on κ. This is guaranteed by the following lemma which states that if there exists a σ-additive measure on a cardinal κ, then there exists a cardinal λ having a λ-additive measure.

Lemma 3.4. The least cardinal ω<κ having a σ-additive measure, is such that every measure on κ is κ-additive. 3.1. ASPECTS OF MEASURABILITY 39

Proof. Suppose m is a measure on κ which is not κ additive and {Xα : α<γ< κ} is a set of pairwise disjoint subsets of S with the property m( α<γ Xα) 6= α<γ m(Xα). Since m is σ-additive, we can remove the -at most countableS by P3.2- sets with positive measure and get a set {Yα : α<γ} where m(Yα)=0 and 0

Deﬁnition 3.6. An Ulam (λ, µ) matrix is a collection {Aα,ξ : α<λ, ξ<µ} of subsets of λ, with the properties:

(i) α 6= β → ∀ξ<µAα,ξ ∩ Aβ,ξ = ∅;

(ii) ∀α < λ |λ \ ξ<µ Aα,ξ|≤ µ. S Theorem 3.7. (Ulam) An Ulam (λ+, λ) matrix exists. Proof. Since λ+ is the successor cardinal of λ, there is a family of functions + + fα : λ → λ , α<λ such that α ⊂ rang(fα). We let Aα,ξ = {γ : fγ (ξ)= α}, hence

• Aα,ξ ∩ Aβ,ξ = {γ : α = fγ(ξ)= β} = ∅ if α 6= β;

+ + • |λ \ ξ<µ Aα,ξ| = |{β : α∈ / rng(fβ)}| < λ .

S + Thus, the set containing Aα,ξ, α<λ, ξ<µ is a (λ , λ) Ulam matrix. Corollary 3.8. Every real-valued measurable cardinal κ is weakly inaccessible. Proof. Suppose m is a κ-additive measure on κ. First of all, we should notice that if X ⊂ κ ∧ |X| <κ, then

m(X)= m( {{x} : x ∈ X})= m({x})=0. [ xX∈X

If κ is not regular then κ = α<µ λα, µ,λα < κ thus |κ| = α<µ |λα| = 0 which is a contradiction. P P 40CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

Suppose that κ = λ+. Let us consider an Ulam (λ+, λ) matrix, just as the one in the above theorem. By (ii) of theorem 3.7, for every α < λ there is + some ξα such that m(Aα,ξα ) > 0. Since λ < λ there is a ξ for which there are + λ pairwise disjoint sets Aα,ξ of the matrix, such that |Aα,ξ| > 0, which is a contradiction by lemma 3.2. Hence, κ must be a limit cardinal.

Atomic measures In order to deal with the case where an atomless measure exists, we quote a lemma concerning partition properties of atomless measures and afterwards a theorem, which clariﬁes this case. Lemma 3.9. Let m be an atomless κ-additive measure over κ. Then: (i) For any ǫ ∈ [0, 1] and X ⊂ κ, 0 < m(X) there is a Y ⊂ X such that 0

(ii) We deﬁne a sequence Xα,α<ω1 as follows:

– X0 = X; 1 – 2 m(X)

– Xα = β<α Xβ if limord(α). T It is easy to see by induction that every Xα has measure greater or equal 1 1 to 2 m(X). If those measures were all greater than 2 m(X), then the set {Xα \ Xα+1 : α <ω1} would be an uncountable collection of sets of nonzero measure, which contradicts lemma 3.2.

Theorem 3.10. (Ulam) If there exists an atomless κ-additive measure m over κ, then:

(i) κ ≤ 2ℵ0 ; (ii) there is a measure µ over R extending Lebesque’s measure (for a deﬁnition and more information on measure theory see [16]). Proof. 3.1. ASPECTS OF MEASURABILITY 41

(i) The idea behind this proof is that the partition property of lemma 3.9 (ii) leads to the construction of a partition of κ into 2ℵ0 sets of measure 0, which means that m is not (2ℵ0 )+-additive, thus κ ≤ 2ℵ0 . <ω The sets Xs ⊂ κ, s ∈ ω are the ones who generate this partition. We deﬁne

– X∅ = κ; 1 – Xs = n<ω Xs⌢hni, m(Xs⌢hni)= 2i+1 m(Xs); (partitionS property (ii) applied ω times)

where the sets Xs⌢hni, n<ω are pairwise disjoint. Finally, we define ω Xf = n<ω Xf↾n, f ∈ ω and it is straightforward that m(Xf ) = 0, 1 becauseT for every n<ωm(Xf ) < 2n . ∗ ω (ii) We first define the measure m on A = {Xf : f ∈ ω } with the property m∗(Y ) = m( Y ) for every Y ⊂ A. The transformation F : A → Z [k, k + 1], k ∈ ,S where F (Xf ) = k + n<ω m(Xf↾n), is one to one, onto [k, k + 1] and thus [k, k + 1] inherits,P in a natural way, a measure µ[k,k+1]. Those measures, for different k ∈ Z, generate the measure µ, R µ(X)= k∈Z m(X ∩ [k, k + 1]),X ⊂ . µ can be easily checked to be an extensionP of Lebesque’s measure.

If we consider that κ, in the above theorem is weakly inaccessible, we come to the conclusion that the existence of an atomless measure results to the failure of the CH. Even more, the following holds:

Theorem 3.11. (Banach-Kuratowski) CH → there is no measure on 2ℵ0 .

Proof. See [8].

Two valued measures We now turn to the case of two-valued measures.

Lemma 3.12. A two-valued κ-additive measure m on a set S deﬁnes a κ- complete nonprincipal ultraﬁlter U such that:

U = {X ⊂ S : m(X)=1}.

Conversely, a κ-complete nonprincipal ultraﬁlter U on S deﬁnes a two-valued κ-additive measure m on S such that: 0 X/∈ U m(X)= 1 X ∈ U

Proof. It is straightforward from the corresponding deﬁnitions. 42CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

If the measure is also real valued on κ, it deﬁnes a κ-complete nonprincipal ultraﬁlter on κ. This way the notion of measurable cardinals, which plays an important role on the theory of large cardinals, turns up.

Deﬁnition 3.13. A cardinal ω<κ, is called measurable if there exists a κ- complete nonprincipal ultraﬁlter on κ.

Lemma 3.14. Every measurable cardinal κ is inaccessible.

Proof. κ is also real-valued measurable thus it is regular. Let λ<κ and κ ≤ 2λ. Let f : κ → 2λ be a 1-1 function and U a nonprincipal κ-complete ultraﬁlter on κ. Then, there is a sequence of sets Xα and dα ∈ {0, 1}, α < λ such that Xα = {ξ<κ : f(ξ)(α) = dα} ∈ U. The cut of those sets belongs to U since it is κ-complete but it has one member which opposes to the nonprincipality of the ultraﬁlter. Hence, κ is strong limit.

Combining the results we have mentioned up to now in this section we get the following theorem:

Theorem 3.15. Suppose there is a σ-additive measure on S. Then, the following are true:

(i) If there exists a set, such that A mA is two-valued, in the partition described in lemma 3.3, then there exists a measurable cardinal.

(ii) If there exists a set B, such that mB is atomless, in the partition mentioned above, then there is a real-valued atomless measure on a weakly inaccessible cardinal κ ≤ 2ℵ0 .

Remark 3.16. We have to notice here that the existence of a real-valued atomless measure is not only inconsistent with the CH but it also implies that the ℵ0 gap between ℵ1 and 2 is wide. On the other hand, the existence of a measurable cardinal does not seem to eﬀect CH, as it is inaccessible thus greater than 2ℵ0 . In fact, using mild forcing extensions, one can prove that assuming the existence of a measurable cardinal the GCH can either be true or false (see [9] or [8]).

3.2 Elementary embeddings of V

The elementary embeddings of V are tightly related to measurable cardinals. In particular, the existence of a nontrivial elementary embedding1 is equivalent to the existence of a measurable cardinal. In this section we are going to prove this result, perform a quick investigation on elementary embeddings and see some of the consequences of their existence. One of those will be the fact that V 6= L if a measurable cardinal exists. This is an important result due to Scott.

1Diﬀerent from the identity function. 3.2. ELEMENTARY EMBEDDINGS OF V 43

Elementary embeddings and ultrapowers of inner models We have defined elementary embeddings and ultrapowers for models which are sets, but here we have to deal with inner models which are classes. The easiest way to redefine those notions, would be by working on a higher order set theory such as Morse-Kelley, but it can still be done in ZF C and thus maintain the simplicity of first order logic. From now on we will work on the extended language L = {∈, j}, where j is a function symbol. We will also add new axioms which state that j is an elementary embedding and that the axiom of replacement holds in the extended language. Lemma 3.17. Let M,N be inner models such that M ≺ N, then: (i) ordinal(α) → ordinal(j(α)) ∧ α ≤ j(α); (ii) if j is nontrivial and N ⊂ M, then ∃δ ∈ On δ

Definition 3.18. The least ordinal α such that α < j(α) will be called the critical point of j. We will now focus our interest in generalizing the technique of ultrapowers to classes, and specifically to V . The difficulty here is in defining the classes of functions so that they are sets. Definition 3.19. Let U be an ultrafilter on S, M an inner model and f,g,h : S → M, then:

(i) f =U g ↔{x ∈ S : f(x)= g(x)} ∈ U;

(ii) f ∈U g →{x ∈ S : f(x) ∈ g(x)} ∈ U;

(iii) [f]= {g : f =U g ∧ ∀h (h =U f → rank(g) ≤ rank(h))}; 2 (iv) Ult(M,U) = ({[f]: f : S → M}, ∈U ). 2 We use the same symbol for inclusion in classes, i.e. [f] ∈U [g] ↔ f ∈U g. 44CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

The following theorems also hold for classes and they are proved the same way as for sets. Theorem 3.20.

(i) For any fomula φ and functions f1,...,fn : S → M,

Ult(M,U) φ[[f1],..., [fn]] ↔{x ∈ S : φ[f1(x),...,fn(x)]} ∈ U

(Thus Ult(M,U) ZF C).

(ii) j : M → Ult(M,U), j(x) = [cx], where cx : S → M is the identity function, is elementary. We call this embedding the canonical embedding.

Elementary embeddings of V Now that we have generalized the notion of elementary embeddings and ultrapowers, we shall introduce our ﬁrst example of a nontrivial elementary embedding j : V ≺ Ult(V,U) and study its properties. Before we do this though, we must make sure that Ult(V,U) is an inner model, or at least that it is isomorphic to one. This will be carried out through the following lemma. Lemma 3.21.

(i) The model Ult(V,U) is extensional.

(ii) For every [f] ∈ Ult(V,U), ext∈U ([f]) is a set. (iii) If U is σ-complete, then Ult(V,U) is also well-founded.

Proof.

(i) Ult(V,U) ZF C thus Ult(V,U) “extensionality axiom”.

(ii) Let [g] ∈U [f]. By setting

g(x) g(x) ∈ f(x) h(x)= ∅ else

we get a function h such that h =U g, and rank(h) ≤ rank(f). Hence

ext∈U ([f]) ∈ Vrank(f)+1, which means that it is a set. (iii) Suppose U is σ-complete and Ult(V,U) is not well-founded. In this case there is an ω-sequence {[fn]}n<ω such that ∀n<ω [fn+1] ∈U [fn]. Hence the sets Xn = {x ∈ S : fn+1(x) ∈ fn(x)} are all members of U, and by σ-completeness n<ω Xn ∈ U → n<ω Xn 6= ∅. If x ∈ n<ω Xn, then there exists an inﬁniteT decreasingT sequence f0(x) ∋ f1(x)T∋ ..., which is a contradiction, thus Ult(V,U) is well-founded. 3.2. ELEMENTARY EMBEDDINGS OF V 45

The above lemma combined with Mostowski’s collapsing lemma produces an inner model MU , isomorphic to Ult(V,U). For our convenience, we will denote the elements of MU by their isomorphic images in Ult(V,U). Theorem 3.22. Suppose there is a measurable cardinal κ and U is a nonpriciple κ-complete ultraﬁlter on κ. Then the canonical embedding j : V ≺ MU is a nontrivial elementary embedding.

Proof. By theorem 3.20 we know j is elementary so we only need to show that it is nontrivial. We will ﬁrst show that ∀α<κj(α) = α and then, using this, we will prove that κ < j(κ).

• j(0) = 0 and ∀α ∈ On j(α +1) = j(α) + 1 by the absoluteness of the corresponding terms. Suppose that α<κ ∧ limord(α) and for every ordinal β less than α, j(β) = β. If [f] < j(α) = [cα] then for almost all x ∈ κ f(x) < α and since α<κ, by the κ-completeness of U, there is a β < α such that f =U cβ. Hence j(α) ≤ α so by lemma 3.17 j(α)= α. • Let id : κ → V, id(α) = α. Since U is κ-complete |S \ x| < κ → x ∈ U, thus α = j(α) = [cα] < [id], α<κ which means that κ ≤ [id]. On the other hand, [id] < [cκ]= j(κ), hence κ < j(κ).

We will now prove the inverse:

Theorem 3.23. (Keisler) If there is a nontrivial elementary embedding j : V ≺ M, then its critical point is a measurable cardinal.

Proof. Suppose κ is the critical point of j. κ is greater than ω so we only have to check that there is a κ-complete nontrivial ultraﬁlter U on κ. Let U = {X ⊂ κ : κ ∈ j(X)}, then:

• j(∅)=0 6∋ κ, κ ∈ j(κ), j(X ∩ Y )= j(X) ∩ j(Y ), X ⊂ Y → j(X) ⊂ j(Y ) and j(κ \ X)= j(κ) \ X by elementarity of j, therefore it is clear that U is an ultraﬁlter.

• If α<κ, then j({α})= {j(α)} = {α}, hence κ∈ / j(α) which means that U is nonprincipal.

• Let α<κ and S = {Xβ}β<α be a sequence of subsets of κ such that for every β, κ ∈ j(Xβ). j(S) will also be a sequence, of length j(α) = α, having the property j(S)(β)= j(Xβ), β<α. Hence

′′ ′′ j( S α)= j(S) α = j(Xβ) \ \ β<α\

which means that κ ∈ j( S′′α), so U is κ-complete. T 46CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

Corollary 3.24. The existence of a nontrivial elementary embedding j : V ≺ M is equivalent to the existence of a measurable cardinal.

Proof. It has been carried out by the two preceding theorems. The only diﬀer- ence here is that, since we want to talk about the existence of an embedding j, we must work in the extended language L∗ = {∈, j} or think of this result as a metatheorem.

Corollary 3.25. (Scott) If there exists a measurable cardinal then V 6= L.

Proof. Suppose κ is the least measurable cardinal, U a κ-complete nonprincipal ultraﬁlter over κ and j : V ≺ MU the elementary embedding described above. If V = L then, since L is contained in every inner model, MU = L = V . j is elementary so V = MU “j(κ) is the least measurable cardinal” which is a contradiction because κ < j(κ).

The last result, due to Scott, yields that the existence of large cardinals can drastically change the structure of V . It was also the ﬁrst result which introduced a connection between elementary embeddings of V and large cardinals. We will now give some more information on the elementary embeddings of the form j : V ≺ MU .

Lemma 3.26. If U is a nonprincipal κ-complete ultraﬁlter on κ and j : V ≺ MU is the canonical elementary embedding, then:

MU (i) – ∀x ∈ Vκ j(x)= x so Vκ = Vκ;

MU + MU + – ∀X ⊂ Vκ j(X) ∩ Vκ = X, thus Vκ+1 = Vκ+1 and (κ ) = κ .

(ii) 2κ ≤ (2κ)MU < j(κ) < (2κ)+.

(iii) For every limit ordinal α:

– cf(α)= κ → limβ→α j(β) < j(α);

– cf(α) 6= κ → limβ→α j(β)= j(α). (iv) If λ is a strong limit cardinal then cfλ 6= κ → j(λ)= λ.

κ (v) – MU ⊂ MU (MU is closed for κ-sequences); κ+1 – MU 6⊂ MU (MU is not closed for κ +1-sequences).

(vi) U/∈ MU . Proof.

(i) – Suppose x ∈ Vκ then, since κ is limit, rank(x) <κ. If j(x) 6= x then we would have j(rank(x)) 6= rank(x), just as in lemma 3.17, which is a contradiction.

ZF MU Vα is Π1 thus (Vκ) = Vκ ∩ MU , so from what we mentioned MU before (Vκ) = Vκ. 3.2. ELEMENTARY EMBEDDINGS OF V 47

– If X ⊂ Vκ then x ∈ X → x = j(x) ∈ j(X) thus X ⊂ j(X) ∩ Vκ. On the other hand, j(x)= x ∈ j(X)∩Vκ → x ∈ X, hence j(X)∩Vκ = X. MU Again (Vκ+1) = Vκ+1 ∩ MU and for all X ⊂ Vκ X = j(X) ∩ MU MU (Vκ) → X ∈ MU thus (Vκ+1) = Vκ+1. ZF MU The term w.o.(x) = {<:< well-orders x} is Π1 so (w.o.(κ)) = MU w.o.(κ) ∩ MU , but w.o.(κ) ⊂ Vκ+1 ∈ MU → (w.o.(κ)) = w.o.(κ). + MU + + MU This means that κ is Σ1 , hence κ ≤ (κ ) . In order to show + + MU that κ = (κ ) we must find an, absolute for MU , function f, + in MU , which shows that w.o.(κ) and κ have the same cardinality. We define by transfinite recursion F (α, x, <) =“the α-th element of hx,

(ii) 2κ = |P (κ)| and, since P (κ)MU = P (κ) (it follows from the fact that MU ZF κ κ MU (Vκ+1) = Vκ+1) and |x| is Π1 , 2 = (2 ) .

κ < j(κ) and j(κ) is measurable, therefore inaccessible in MU , so

(∀α 2α < j(κ))MU → (2κ)MU < j(κ).

κ κ κ + |j(κ)| = |[cκ]| = κ · [κ] =2 < (2 ) .

(iii) – β < α → j(β) < j(α) hence limβ→α j(β) ≤ j(α). If cf(α) = κ, then there is a coﬁnal function f : κ → α.[f] acts similarly to [id], i.e. ∀β<αj(β) < [f] but [f] < j(α) so limβ→α j(β) < j(α). – If cf(α) >κ, then [f] < j(α) implies that for almost every xf(x) < α therefore ∃β < α [f] < j(β). For this reason limβ→α j(β)= j(α). If cf(α) <κ, then for every[f] < j(α) there is again, by κ-completeness, a β < α such that [f] < j(β).

κ (iv) If α < λ, then |[α]| ≤ |α | < λ thus j(λ) = limα→λ j(α) ≤ λ. Since λ ≤ j(λ), j(λ)= λ.

(v) – Let s = h[fα]: α<κi be a κ-sequence, [g] = κ and [h], h(α) = h[fβ]: β

κ ZF κ κ MU (vi) κ is Π1 hence, by (v), ( κ) = κ which belongs to MU . If U also ′′ κ belongs in MU then there is an absolute term F (f) = [f], so F κ ∈ MU κ which is a function onto j(κ). This means that MU |j(κ)| ≤ 2 which is a contradiction since MU “strong limit j(κ)” and MU κ < j(κ). 48CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

From (vi) we see that MU 6= V , thus we cannot define a nontrivial embedding j : V ≺ V using the ultrafilter construction we have described. In the last chapter we will show that under reasonable assumptions no such embedding can exist. Finally, we state the following application of this lemma, which shows that the existence of measurable cardinals has effects on cardinal arithmetic. Corollary 3.27. (Scott) Suppose κ is measurable. Then,

∀α<κ 2α = α+ → 2κ = κ+.

Proof. Let j : V ≺ MU be the corresponding canonical embedding. By elementarity,

α + α + κ MU + MU V ∀α<κ 2 = α → MU ∀α < j(κ) 2 = α → (2 ) = (κ ) .

We already know that 2κ ≤ (2κ)MU and that κ+ = (κ+)MU so 2κ = κ+

3.3 Normal measures, indescribability

We are now going to study normal measures and the connection of measurable cardinals with indescribability and Ramsey cardinals. Definition 3.28. Let κ be a measurable cardinal. A normal filter D on κ is called a normal measure on κ. As we have already mentioned, every normal filter D contains the club filter on κ, thus all of its sets are stationary. We have also seen that a filter is normal if and only if every regressive function on a set X ∈ D is constant on a Y ∈ D, such that Y ⊂ X. We will show that there is a normal measure D for every measurable cardinal and then use this to prove that every measurable cardinal is Ramsey. Theorem 3.29. For every measurable cardinal κ there is a normal measure D on κ.

Proof. Let U be a nonprincipal κ-complete ultraﬁlter on κ and A = {[g] ∈ MU : ∀α<κ [cα] < [g]}. A is a nonempty subset of MU , since [id] ∈ A, thus it has a minimal element [h]. Suppose f ∈ [h] ∧ rng(f) ⊂ κ (this is possible because [h] ≤ [id] ∧ rng(id)= κ) and let D = {X ⊂ κ : f −1′′X ∈ U}. It is straightforward to see that D is a κ-complete ultraﬁlter. For every α<κ, f −1′′{α} ∈/ U because [α] < [f], thus D is nonprincipal. Suppose g : X → κ, X ∈ U is a regressive function. Then h = g ◦ f has the property ∀α ∈ f −1′′X h(α)

Theorem 3.30. Every measurable cardinal is a Ramsey cardinal.

Proof. Let κ be measurable and D a normal measure on κ. We will show that every partition f :[κ]<ω → α for every α<κ has a homogeneous set X ∈ D. In order to do this we will inductively deﬁne the sets Xn, 0

Remark 3.31. We have to notice here that in order to establish the above theorem we have used the axiom of choice. Under other circumstances mea- 3 surable cardinals could be quite small, for example ZF +AD ⊢“ω1 is the least measurable cardinal”.

Another theorem concerning normal measures is the following, which gives equivalent characterizations of normal measures, through the ultraproduct models generated by measurable cardinals and elementary embeddings.

Theorem 3.32. If D is a nonprincipal κ-complete ultraﬁlter over κ, then the following are equivalent:

(i) D is normal;

(ii) κ = [id] (in MU ); (iii) for every X ⊂ κ, X ∈ D ↔ κ ∈ j(X).

Proof.

(i)→(ii) We already know that κ ≤ [id]. If [f] ≤ [id] then f is regressive for almost all α<κ thus, by normality, it is constant for almost all α<κ, therefore [f] = [cα], α<κ. Hence, κ < [id] cannot be the case, which means that κ = [id].

(ii)→(iii) X ∈ D ↔{α : id(α) ∈ X} ∈ D which is equivalent to [id] ∈ [X]= j(X). Thus, if κ = [id], X ∈ D ↔ κ ∈ j(X).

(iii)→(i) Suppose f is a regressive function on a X ∈ D. Then, by elementarity of j, α = j(f)(κ) < κ. This means that κ ∈ j({β<κ : f(β) = α}) so f is equal to α for almost all β<κ.

3The axiom of determinacy, see [10] or [14]. 50CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V

2 Finally, we show that every measurable cardinal is Π1-indescribable and we use this to prove that the least measurable cardinal is greater than the least Ramsey cardinal. We also see some more connections of measurable cardinals with indescribability.

2 Theorem 3.33. Every measurable cardinal is Π1-indescribable.

Proof. Let φ = ∀X ψ(X), where ψ(X) is a second order formula, R ⊂ Vκ and (Vκ, ∈, R) φ. By the truth deﬁnition, we have that ∀X ⊂ Vκ+1 (Vκ+1, ∈ Vκ MD MD ,X,Vκ, R) ψ . Since (Vκ+1) = Vκ+1 and (Vκ) = Vκ, ((Vκ, ∈,U) MD MD φ) and furthermore ((Vκ, ∈, j(R) ∩ Vκ) φ) because R = j(R) ∩ Vκ. If A = {α<κ : (Vα, ∈, R ∩ Vα) φ} then, by elementarity of j, κ ∈ j(A), hence by normality of D, A ∈ D → A 6= ∅.

The above result is in some way the best possible since:

2 Theorem 3.34. The least measurable cardinal is not Σ1-indescribable.

2 Proof. There is a sentence φ ∈ Σ1 such that (Vκ, ∈) φ ↔ measurable(κ):

∃U ∈ P P (Vκ) (∀X ∈ P (Vκ) (X ∈ U → X ⊂ On ∧ ∅ ∈ U ∧ On ∈ U∧

∧ ∀X, Y ∈ P (Vκ) X ∩ Y ∈ U∧

∀X, Y ∈ P (Vκ) (X ∈ U ∧ X ⊂ Y → Y ∈ U)∧

∧ ∀X ∈ P (Vκ) (X ∈ U ∨ On \ X ∈ U)∧ ′′ ∧ ∀F : Vκ → Vκ∀α ∈ On F α ∈ U). \ Hence the least measurable cardinal κ is such that (Vκ, ∈) φ but for all α<κ, 2 (Vα, ∈) 6 φ. Thus, it is not Σ1-indescribable.

2 Though the indescribability of measurable cardinals is up to Π1, there are many totally indescribable cardinals below every measurable cardinal. In particular:

Theorem 3.35. If κ is measurable and D is a normal measure on κ, then {λ<κ : λ is totally indescribable} ∈ D.

Proof. As we have seen in the previous proofs, we just have to show that MD (κ is totally indescribable) for a normal measure D. Suppose (R ⊂ Vκ ∧ MD m (Vκ, ∈, R) φ) , φ ∈ n,m<ω Πn . Then, S MD (∃α < j(κ) (Vα, ∈, j(R) ∩ Vα) φ) →

∃α<κ (Vα, ∈, R ∩ Vα) φ →

MD (∃α<κ (Vα, ∈, R ∩ Vα) φ) . 3.3. NORMAL MEASURES, INDESCRIBABILITY 51

Remark 3.36. The latter reveals the fact that indescribability is not enough to support the existence of measurable or greater large cardinals (provided that AC holds). Thus the elementary embeddings of V take its place as a stronger reﬂection principle, used to produce large cardinals. In the ﬁfth chapter we will encounter more examples of such embeddings and large cardinals related to them.

2 Using Π1-indescribability we prove the next corollary: Corollary 3.37. If D is a normal measure over the cardinal κ, then {λ<κ : Ramsey(λ)} ∈ D.

2 Proof. As in 3.34, we can ﬁnd a Π1 sentence φ such that Ramsey(λ) ↔ (Vλ, ∈ ) φ. Hence the assertion follows from the proof of 3.30. We close this section having a quick view at the models of the form L[D], where D is a normal measure.

Theorem 3.38. (Silver) If D is a normal measure on a cardinal κ, then L[D] GCH. Theorem 3.39. (Kunen) If D is a normal measure on a cardinal κ and V = L[D], then κ is the only measurable cardinal and D the only measure on κ. 52CHAPTER 3. MEASURABLE CARDINALS AND ELEMENTARY EMBEDDINGS OF V Chapter 4

0♯ and elementary embeddings of L

4.1 Indiscernibles

Before we proceed further in this chapter, we first need to introduce one more model theoretical notion, indiscernibility. A set I is called a set of indiscernibles, if we cannot distinguish its elements using first order formulas. Thus, if one formula is satisfied by a n-tuple of In, then it is satisfied by all the other n- tuples of In. Since the sets of indiscernibles we will consider shall contain ordinals, which come along with a natural well-ordering, we will require that the n-tuples are entered in an increasing order in every formula we will consider. Finally, after introducing indiscernibility, we will see how new models can be created by extending sets of indiscernibles, using Skolem functions, and state some of their properties. Definition 4.1.

(i) Let M be a structure and (I,<) a linearly ordered set. I is a set of indiscernibles for M, if for every formula φ(v1,...,vn) and all x1 <...< xn,y1 <...

M φ[x1,...,xn] ↔ M φ[y1,...,yn].

(ii) If M is a structure and I a set of indiscernibles for M, then we denote by Σ(M, I) the set of formulas true in M for increasing sequences of I. A very useful theorem, concerning models which contain sets of indiscernibles, is the following: Theorem 4.2. (Ehrenfeucht-Mostowski) If T is a theory with inﬁnite models, and (I,<) is a linearly ordered set, then there exists a model M of T such that (I,<) is a set of indiscernibles in M.

53 54 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

Proof. It can be found in [2], [10] or [7] and it is an interesting application, of Ramsey’s theorem, which gives us a connection of inﬁnitary combinatorics with model theory.

The following theorem provides us with the basic properties of models constructed from indiscernibles, and its proof can be found in [2].

Theorem 4.3. Let M be a model, of a theory T with built in functions, and (I,<) a set of indiscernibles of M. Then,

(i) If J ⊂ I, then J with the inherited order is a set of indiscernibles in H(J) and H(J) ≺ H(I).

(ii) If I, J are inﬁnite linearly ordered sets, then there is a model N of T , such that J is a set of indiscernibles for N and Σ(M, I)=Σ(N,J).

(iii) Let N be a model of T and J a set of indiscernibles for N such that Σ(M, I) = Σ(N,J). Every increasing function j : I → J can be uniquely extended to an elementary embedding ¯j : H(I) → H(J). Moreover, ¯j′′H(I)= H(j′′I).

The theorems above are the starting point of an attempt of producing models with many automorphisms, initiated by Ehrenfeucht and Mostowski. By theorem 4.3 (iii), if j : I → I is an increasing onto I function then it can be extended to an automorphism of H(I). Thus, if a model is produced by a set of indiscernibles, with quite a flexible order1, then it has a lot of automorphisms. The way we are going to use those theorems is much alike, i.e. we wish to verify the existence of a non-trivial elementary embedding j : L → L. Of course, this comes with a cost; we must make stronger combinatorial assumptions, not provable in ZF C, which will guarantee the existence of appropriate sets of indiscernibles. This was first done using the existence of Ramsey cardinals and through this procedure it became clear that a weaker assumption, the existence of 0♯, is enough. 0♯ is a Σ(M, I) set of formulas, having some specific properties.

4.2 0♯

Near the sixties, it had already been noticed, by set theorists, that the existence of several large cardinals has a great impact on the structure of L relative to V . We have already seen in the previous chapter that the existence of a measurable cardinal implies that V 6= L. This impact could be described as a compression of L and it is connected to results such as |L ∩ R| = ℵ0, L inaccessible(ℵα) for every α > 0 and the existence of elementary embeddings of L. 0♯ emerged exactly from the study of those phenomena.

1One which has many <-automorphisms. 4.2. 0♯ 55

E-M sets 0♯’s deﬁnition is not straightforward. In order to understand its nature, one must go through the procedure of its creation. This is why we ﬁrst introduce the E- M sets, which are related to the theory of indiscernibles. Roughly speaking, in order to shrink L we must make sure that its structure is simple, which can be done by assuming that many of its elements have the same properties or, in other words, that it contains an appropriately large set of indiscernibles. From now on, the sets of indiscernibles for the models M we will consider, will contain only ordinals of M (elements of OnM ), ordered by ∈M .

Deﬁnition 4.4. Let

the < −least y such that φ(y, x¯) h (¯x)= L φ ∅ otherwise

Lemma 4.5. Let α, β be limit ordinals, j : Lα → Lβ an elementary embedding and φ a formula. If x1,...,xn ∈ Lα, then

Lβ Lα hφ (j(x1),...,j(xn)) = j(hφ (x1,...,xn)).

Proof. We may notice that the formula x

Lβ Lβ the

Lα =j(hφ (x1,...,xn))

In order to define 0♯ we introduce the following sets: Definition 4.6. We will call a set of formulas Σ an E-M set (Ehrenfeucht- Mostowski), if there is a model M, an infinite set I of indiscernibles for M and a limit ordinal α such that:

(i) M ≡ Lα (ii) Σ = Σ(M, I). The following theorem is a special case of theorem 4.2: Theorem 4.7. If Σ is an E-M set and α an inﬁnite ordinal, then there is a unique up to isomorphism pair (M, I), where M is a model of Σ and I a set of indiscernibles for M having the following properties: (i) Σ=Σ(M, I); 56 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

(ii) the order type of I is α;

(iii) M = H(I).

Proof.

(∃) Since Σ is an E-M set, there is a model N and an inﬁnite set of indiscernibles J for N such that Σ = Σ(N,J). Let {jn : n<ω} be a countable subset of J. By theorem 4.3 (ii), there exists a model M ′ and a set of indiscernibles I for it with order type α, such that Σ = Σ(N,J)=Σ(M ′, I). The submodel M = H(I) of M ′, is exactly what we need, since it is an elementary submodel of M ′ thus I is a set of indiscernibles for it, and Σ(M, I)=Σ(M ′, I) = Σ.

(∃!) The uniqueness of the pair (M, I) is immediate by theorem 4.3 (iii). If (N,J) is another pair, then there is an <-isomorphism, j : I → J which can be extended to an isomorphism between M and N.

Deﬁnition 4.8. We will call the unique pair (M, I), of theorem 4.7, the (Σ, α)- model.

It is immediate from theorem 4.3 (iii), that

α ≤ β → (Σ, α) ≺ (Σ,β).

We are now going to analyze three properties related to (Σ, α) models. Speciﬁcally we are going to consider the cases where a (Σ, α) model is well- founded, unbounded, or remarkable. These properties are used in order to identify some of the (Σ,κ) models, cardinal(κ), κ>ω, with the models Lκ. This way the models Lκ and L will be built up from sets of indiscernibles, thus we will have the opportunity of getting the results we have already described.

Deﬁnition 4.9. Let (M, I) be a (Σ, α)-model and I = {iβ : β ∈ α}, α<β ↔ iα

(ii) (Σ, α) is remarkable if it is unbounded and M (x ∈ On∧x

The following lemma provides us with expressions equivalent to the properties described above: 4.2. 0♯ 57

Lemma 4.11. Let Σ be an E-M set. Then, (a) The following are equivalent: (i) For every ordinal α, (Σ, α) is well-founded.

(ii) For some ordinal α ≥ ω1, (Σ, α) is well-founded.

(iii) For every ordinal α<ω1, (Σ, α) is well-founded. (b) The following are equivalent: (i) For every ordinal α, (Σ, α) is unbounded. (ii) For some ordinal α, (Σ, α) is unbounded. (iii) For every Skolem term t(¯v),

[t(¯v) ∈ On → t(¯v)

(c) The following are equivalent if Σ is unbounded: (i) For every ordinal α>ω, (Σ, α) is remarkable. (ii) For some α>ω, (Σ, α) is remarkable. (iii) For every Skolem term t(¯v, u¯),

[t(¯v, u¯)

(d) If (M, I) is a remarkable (Σ, α) set, then for every limit ordinal β < α, M M (x ∈ On ∧ x

Definitions of 0♯ We are nearly ready to define 0♯. It will be the unique well-founded, remarkable E-M set, but we have not yet verified this uniqueness. In order to do this, we will first prove that the existence of E-M sets of this kind have two important consequences stated in 4.12. Later on, those properties will lead us to a second definition of 0♯. Theorem 4.12. (Silver) The existence of a well-founded, remarkable E-M set has the following consequences:

(i) For all cardinals ω<κ<λ, (Lκ, ∈) ≺ (Lλ, ∈). (ii) There is a unique club class I in On, containing all cardinals and such that for every ω<κ: – |I ∩ κ| = κ 58 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

– I ∩ κ is a set of indiscernibles for Lκ

– ∀α ∈ Lκ, α is definable in Lκ from I ∩ κ. Proof. We will carry out the proof using the following lemmas. In every case we will assume that Σ is a well-founded, remarkable E-M set. Lemma 4.13. If κ>ω is a cardinal, then the (Σ,κ) model is isomorphic to Lκ. Proof. By checking the proof of Gödel’s condensation lemma, we can notice that since Σ is an E-M set, it satisfies φL thus the (Σ,κ)-model, being well-founded, is isomorphic to (Lα, I) for some ordinal α. |I| = κ so κ ≤ α. Suppose κ < α, then since (Σ,κ) is unbounded there is a β<κ such that Lα κ

Lα Lα γ ∈ H ({iδ : δ<β}) → κ ⊂ H ({iδ : δ<β}).

Lα The last inclusion is false, because |H ({iδ : δ<β})| = β<κ, and as a result of that β = κ. Relying on the above lemma, if Σ is well-founded and remarkable we can identify the (Σ,κ)-model with (Lκ, Iκ). Thus if ω<κ<λ, we have that (Σ,κ) ≺ (Σ, λ), which means that Lκ ≺ Lλ and this settles (i) of theorem 4.12. Lemma 4.14. If ω<κ<λ, then

(i) Iλ ∩ κ = Iκ

L (ii) H λ (Iκ)= Lκ. Proof.

L (i) Let J = {iα : α<κ}⊂ Iλ. Then, (H λ (J),J) is a (Σ,κ)-model hence ,by the above lemma it, is isomorphic to (Lκ, Iκ). By lemma 4.11 (d) and (e), L we can see that the ordinals of H λ (J) are exactly iκ. This means that iκ = κ and J = Iκ, thus

Iλ ∩ κ = J = Iκ.

Lλ (ii) We already know that (H (Iκ), Iκ) =∼ (Lκ, Iκ). Thus the isomorpism L of H λ (Iκ) and Lκ is the identity function on Iκ. Provided that every L Skolem function is uniquely deﬁned, by

This lemma proves the second part of theorem 4.12, except from the uniqueness of I. In particular, if we set I = ω<κ Iκ, then: S • |I ∩ κ| = |Iκ| = κ

• I ∩ κ = Iκ which is a set of indiscernibles for Lκ 4.2. 0♯ 59

• Lκ = H(Iκ) so every element of Lκ is deﬁnable in Lκ from Iκ = I ∩ κ. • I ∩ κ is unbounded for any cardinal κ so, since it is closed, κ ∈ I. The next two lemmas are used to prove the uniqueness of I. Lemma 4.15. There is at most one well-founded remarkable E-M set.

Proof. We already know that {ℵn : n ∈ ω} ⊂ I. If Σ is well-founded and remarkable, then Lℵω is the (Σ, ℵω)-model hence for every formula φ

φ(v1,...,vn) ∈ Σ ↔ Lℵω φ[ℵ1,..., ℵn]. Thus Σ is unique. Lemma 4.16. For every regular κ>ω there is at most one club set of indiscernibles Iκ for Lκ having the property Lκ = H(Iκ). Proof. Let J be another set of indiscernibles having the above properties. Since both J and Iκ are club their cut is also club thus inﬁnite. Provided that they are both sets of indiscernibles, this means that

Σ(Lκ,J)=Σ(Lκ,J ∩ Iκ)=Σ(Lκ, Iκ) hence (Lκ,J) is the (Σ,κ)-model so J = Iκ.

It follows from the lemma above, that I is unique because each of the Iκ is unique. Thus we have completed the proof of theorem 4.12. By lemma 4.15 we can define 0♯: Definition 4.17. 0♯ is the unique well-ordered remarkable E-M set. The equivalent definition below is immediate from the proof of 4.15: Corollary 4.18. If 0♯ exists then,

♯ 0 = {φ ∈ Form : Lℵω φ[ℵ1,..., ℵn]}. Another deﬁnition is the following: Corollary 4.19. If 0♯ exists then,

♯ 0 = {φ ∈ Form : L φ[ℵ1,..., ℵn]}. Proof. It follows from the reﬂection principle that for every formula φ there is a cardinal κ> ℵω such that for all x1,...,xn ∈ Lκ

L φ[x1,...,xn] ↔ Lκ φ[x1,...,xn].

By letting x1,...,xn be ℵ1,..., ℵn and considering that Lℵω ≺ Lκ, we have that

Lκ φ[ℵ1,..., ℵn] ↔ Lℵω φ[ℵ1,..., ℵn] ♯ thus if we replace Lω by L in corollary 4.18, we get the same set, i.e. 0 . 60 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

If we replace the formulas by their G¨odel numbers, we can view 0♯ as a set of integers. It can also be considered as a real number and there are many more equivalent deﬁnitions. The one in the corollary above though, is probably the most common one. Finally, we have to state that 0♯ may not be a large cardinal, but its existence ♯ implies the existence of ηα, for every α<ω1, and if ηω+1 exists, then 0 also exists. Hence, its existence can be regarded to be a large cardinal axiom.

4.3 The connection of 0♯ with L

As we have already mentioned, the existence of 0♯ is related to the diﬀerence between the structure of V and L. Here we will analyze this relation in two ways. First we will see that the existence of 0♯ implies that V 6= L. It also has some more consequences which show that V is much larger than L assuming this existence. Afterwards, we shall examine what happens in the case 0♯ does not exist. This time we do not have a result of the form V = L2, but Jensen’s covering theorem will guarantee that L is really close to V . Most of the proofs here are omitted and can be found in [8], [10] and [5].

Assuming that 0♯ exists, The next theorems are all corollaries of Silver’s theorem.

Corollary 4.20.

(i) There exists a subset of Form containing exactly Th(L).

(ii) Lκ ≺ L.

(iii) I is a set of indiscernibles for L.

(iv) defL(I)= L.

Proof.

(i) By corollary 4.19 this set is {φ ∈ 0♯ : sentence(φ)}.

(ii) By the reﬂection principle, for every formula φ there is an uncountable λ such that φ is absolute for Lλ, L. We know from Silver’s theorem that either Lκ ≺ Lλ or Lλ ≺ Lκ, depending on which of κ, λ is bigger. Thus, φ is also absolute for Lκ, Lλ which means that it is absolute for Lκ, L too. Since this is true for every φ, Lκ ≺ L.

(iii) This is immediate from the fact that I ∩ κ is a set of indiscernibles for Lκ, for every cardinal κ, and from (ii).

2It would be impossible for a large cardinal axiom to be equivalent with V 6= L, because then its consistency relative to ZF C would be provable in ZF C 4.3. THE CONNECTION OF 0♯ WITH L 61

(iv) If x ∈ L then there is an cardinal κ such that x ∈ Lκ, so x is deﬁnable in Lκ by I ∩ κ. Hence, by (ii), x is also deﬁnable in L by I ∩ κ, thus also from I.

Corollary 4.21.

∀x ∈ L (x ∈ defL(∅) → |x|≤ ω).

Proof. Suppose x is deﬁned in L by a formula φ(v). In this case, the formula

ψ(u)= ∀v (v ∈ u ↔ φ(v)) is such that only x satisﬁes it in L. Since Lℵ1 ≺ L,

L ∃!u ψ(u) → Lℵ1 ∃!u ψ(u)

therefore there must be a y ∈ Lℵ1 satisfying ψ which ought to be equal to x.

Thus, x ∈ Lℵ1 which means that x is countable.

By the lemma above we have that no uncountable cardinals can be deﬁned in L. As we will see below, those cardinals apart from being undeﬁnable are also inaccessible in L.

Corollary 4.22. Every uncountable cardinal is inaccessible in L.

Proof. We will use the indiscernibility of uncountable cardinals here, in order to transfer properties from speciﬁc cardinals to all uncountable cardinals. L regular(ℵ1) and also L limcard(ℵω), thus every uncountable cardinal has those properties in L, i.e. every uncountable cardinal is inaccessible in L.

♯ Remark 4.23. Since the existence of 0 implies the existence of ηω, we know from the second chapter that ηω ∈ L. Therefore, using the technique of the lemma above, we can see that every uncountable cardinal in L is Hω and thus it is also Mahlo and weakly compact.

Corollary 4.24. |R ∩ L|≤ ω1. Furthermore, |Vα ∩ L| ≤ |α|.

Proof. Vα ∩ L ∈ defL({α}) so, by indiscernibility, Vα ∩ L ∈ defLκ ({α}) for + κ = |α| . As we did before, we can ﬁnd a formula ψ(v, α) such that Vα ∩ L is + the only set satisfying ψ and so Vα ∩L ∈ Lκ. Hence, |Vα ∩L| < |Lκ| = |α| .

The above corollary was ﬁrst proved by Rowbottom for the case where a Ramsey cardinal exists. This is a stronger result which shows that 0♯ is closer to the collapse of L caused by the existence of speciﬁc large cardinals. Finally, using any of the corollaries 4.21 to 4.24, one can prove that V 6= L.

Theorem 4.25. If 0♯ exists then V 6= L. 62 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

Assuming that 0♯ does not exist, The most important result towards this direction is Jensen’s covering lemma. Everything else we mention afterwards follows from it. Theorem 4.26. (Jensen’s covering theorem) 0♯ does not exist iﬀ for every uncountable set X of ordinals there is a set Y ∈ L such that X ⊂ Y and |X| = |Y |. Proof. See [8].

♯ An example of a set we cannot cover if 0 exists is X = {ℵn : n ∈ ω}. This is because ℵω is uncountable, thus regular in L and sup(X)= ℵω so X cannot be covered by a set Y ∈ L of cardinality less than ℵω. This way, if we also take ♯ under consideration that the existence of 0 implies that ℵω is regular in L, we ♯ see that the existence of 0 is equivalent to the proposition L regular(ℵω). The following three corollaries demonstrate some of the consequences of the non-existence of 0♯.

♯ Corollary 4.27. Suppose 0 does not exist. Then, for every ordinal α ≥ ℵ2, such that L regular(α), cf(α)= |α|. Proof. Let X be a coﬁnal subset of λ and Y a constructible set covering X (or an uncountable set X′, X ⊂ X′ ⊂ λ, if X is countable). Since both Y and λ belong to L, Z = Y ∩ λ ∈ L and it also covers X, i.e. X ⊂ Z ⊂ λ and |Z| = |X| + ℵ1 = cf(λ)+ ℵ1. Z is unbounded in λ and, provided that L regular(λ), |Z| = |λ|. Hence |λ| = cf(λ)+ ℵ1 = cf(λ) because λ ≥ℵ2. Corollary 4.28. If 0♯ does not exist, then singular(κ) → L singular(κ).

Proof. If κ is singular and L regular(κ) then κ ≥ ℵ2 so by corollary 4.27 cf(κ)= κ, which is a contradiction. Corollary 4.29. If 0♯ does not exist then singular(κ) → (κ+)L = κ+. Proof. Suppose α = (κ+)L. κ ≤ α ≤ κ+ thus if α 6= κ+, then |α| = κ. This means that cf(α) < |α| since κ is singular but this is a contradiction since L regular((κ+)L) and by corollary 4.27 cf(α)= |α|.

4.4 Elementary embeddings of L

The main purpose of this section is to present Kunen’s theorem, which states that the existence of 0♯ is equivalent to the existence of a nontrivial elementary embedding j : L ≺ L. This looks like the case of the nontrivial elementary embeddings of V , which exist exactly when a measurable cardinal exists. In fact the last one is a stronger kind of elementary embedding, because if j : V ≺ M is nontrivial then j ↾ L : L ≺ L is also nontrivial. Since L is contained in every inner model M,N, if there is a nontrivial embedding j : M ≺ N then there is also a nontrivial embedding j′ : L ≺ L. Therefore, the weakest assumption one 4.4. ELEMENTARY EMBEDDINGS OF L 63 can make on the existence of nontrivial elementary embeddings of inner models, is the existence of a nontrivial elementary embedding of L. Theorem 4.30. (Kunen) 0♯ exists iff there is a nontrivial elementary embedding j : L ≺ L. Proof. (→) We have proved the existence of a set I of indiscernibles in L, such that L = H(I). By theorem 4.3, every order-preserving function j : I → I can be extended to an elementary embedding ¯j : L ≺ L which is nontrivial if j is nontrivial. There are many such functions j, therefore there also many nontrivial elementary embeddings of L (for example take j(iα)= j(iα+1)). (←) This direction needs more work which will be carried out into three steps. In the first step, we will introduce a variant of the ultraproduct construction for nontrivial embeddings of L. This way using a nontrivial embedding L ≺ L we will define an inner model MD =∼ L and a new nontrivial elementary embedding jD : L ≺ L. In the second step, we study the properties of jD and of some of its reductions. Using those reductions we come up with a set of indiscernibles {γα : α<ω1} for Lκ, where κ is an appropriate cardinal. In the third step, we see that the existence of an uncountable set of indiscernibles for a model Lκ implies the existence of 0♯, which completes the proof.

Step 1 We define below the notions of ultrafilters and ultrapowers, relativized to a model M. Definition 4.31. Suppose M is a transitive model and κ ∈ M. A M-ultrafilter on κ is a subset D of P M (κ) having the following properties: (i) ∅ ∈/ D ∧ κ ∈ D; (ii) X, Y ∈ D → X ∩ Y ∈ D; (iii) X ∈ D ∧ Y ∈ M ∧ X ⊂ Y → Y ∈ D; (iv) ∀X ∈ P M (κ) (X ∈ D ∨ κ \ X ∈ D). D is called nonprincipal if: (v) ∀α ∈ κ {α} ∈/ D. D is called κ-complete if:

(vi) If α<κ, {Xβ : β < α} ∈ M and ∀β<αXβ ∈ D, then β<α Xβ ∈ D. S D is called normal if: 64 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

(vii) Every regressive function f ∈ M on a X ∈ D is constant on a Y ∈ D.

The ultrapower MD of M by D is remains the same as the one defined for ultrafilters and we should only have in mind that D is a M-ultrafilter. We also note by jD the canonical embedding of M in MD. The following theorems we have provided for ultrapowers and elementary embeddings, still hold for M-ultrapowers and they are proved the same way: Theorem 4.32. Suppose D is an M-ultrafilter on κ. Then for every formula φ: MU φ([f1],..., [fn]) ↔{α<κ : M φ(f1(α),...,fn(α))} ∈ D. Theorem 4.33. Let j : M → N be an elementary embedding having κ as its critical point. Then, M (regular(κ) ∧ κ> (ω)M ) and D = {X ∈ P M (κ): κ ∈ j(X)} is a normal nonprincipal κ-complete M-ultrafilter on κ. Suppose j : M ≺ N is an elementary embedding, κ is a critical point of j and D is is the M-ultrafilter described above. It is not hard to check that the function k : MD → N, k([f]) = j(f)(κ), is an elementary embedding such that k ◦ jD = j. This way, we can see that MD is well-founded, since an infinite decreasing sequence {[fn]}n<ω of MD would be mapped to an infinite decreasing sequence {k([fn])} of N. In this case, as in chapter 3, we will identify MD with its transitive collapse which is an inner model. Now we will focus our interest on elementary embeddings j : L ≺ L. LD = L because using theorem 4.32 we can show that LD V = L. From this point on, we will only work with jD : L ≺ L and symbolize its critical point by γ for convenience (κ will be used as an appropriate cardinal for which Lκ has an uncountable set of indiscernibles) .

Step 2

Lemma 4.34. If κ is a limit cardinal such that cf(κ) >γ, then jD(κ)= κ.

Proof. As in the previous chapter, we can check that jD(κ) = limα→κ jD(α). γ L γ L Additionally, for every α<κ, |jD(α)| = |[cα]| ≤ |(α ) | and |(α ) | < κ (because κ is limit so there is a cardinal α<λ<κ for which αγ ≤ λγ = λ<κ). Thus, jD(κ) ≤ κ → jD(κ)= κ. We deﬁne by transﬁnite induction the following decreasing sequence of sets:

U0 ={κ : cardinal(κ) ∧ cf(κ) >γ}

Uα+1 ={κ ∈ Uα : |Uα ∩ κ| = κ}

Uα = Uβ if limord(α) β<α\

It is easy to prove by induction that every class Uα is unbounded and closed for sequences of length β such that cf(β) > κ, thus Uω1 is nonempty. We are going to use an element κ of Uω1 in order to establish our results. 4.4. ELEMENTARY EMBEDDINGS OF L 65

We reduce the mapping jD : L ≺ L we have described before, to i = j ↾ Lκ, which is an elementary embedding of Lκ. We deﬁne the sets Xα = Uα ∩ κ, α<ω1, which have cardinality κ, and afterwards the following elementary submodels of Lκ: Mα = H(γ ∪ Xα).

Suppose Fα is the transitive collapse of Mα. Then, provided that |γ ∪ Xα| = κ, ′′ −1 Fα Mα = Lκ by G¨odel’s condensation lemma. Let i : Lκ ≺ Lκ, iα = Fα , and γα = iα(γ). Our goal is to prove that the ordinals γα, α<ω1, form a set of indiscernibles for Lκ. In order to do this, we are going to create a family of elementary embeddings iα,β of Lκ, which map γα to γβ. In the next two lemmas we give some properties of the ordinals γα and deﬁne the embeddings iα,β. Lemma 4.35.

(i) γα is the least ordinal of Mα greater than γ.

(ii) If α<β then iα ↾ Mβ is the identity function, thus iα(γβ)= γβ.

(iii) α<β → γα <γβ. Proof.

(i) Since γ ⊂ M, γα = iα(γ) is the smallest ordinal of M containing all

the elements of γ, thus supMα (γ). We will show that γ∈ / Mα, therefore iα 6= γ. The elements of Mα are of the form t[δ1,...,δn] (t is a Skolem term), where the ordinals δ1,...,δn belong either to γ or Xα, hence they are not changed by the embedding i. This means that

i(t[δ1,...,δn]) = t[i(δ1),...,i(δn)] = t[δ1,...,δn]

so γ∈ / Mα because i(γ) 6= γ.

(ii) Since the elements of Mβ are of the form t[δ1,...,δn], we have to show that the elements of γ and Xβ are preserved by iα. For γ, it is straightforward because γ ⊂ Mα. As for Xβ, we can see by transﬁnite induction that iα(λ)= λ, since

α<β → |Xα ∩ λ| = λ → Fα(λ) = sup{Fα(µ): µ < λ}. Mα

(iii) It is immediate that γα ≤ γβ, so we only need to chck that γα 6= γβ. This is also true though, since

γ<γα → γα = iα(γ)

Lemma 4.36. For all α<β<ω1 there are elementary embeddings iα,βLκ ≺ Lκ, which preserve all ordinals of ω1 that are not in the interval [α, β] and map γα to γβ. 66 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L

Proof. The embeddings iα,β are deﬁned in a similar way to iα. Speciﬁcally, we −1 let Mα,β = H(γα,Xβ) and iα,β = Fα,β , where Fα,β is the transitive collapse of Mα,β. γα ⊂ Mα,β, therefore iα,β(γξ)= γξ for all ξ < α. By repeating the procedure in lemma 4.35 for Xβ we get that iα,β(γξ)= γξ for all ξ>β. What is left now is to prove that iα,β(γα) = γβ. Similarly to lemma 4.35, γα ≤ iα,β(γα) ≤ γβ. Suppose there is an ordinal ζ ∈ Mα,β, such that γα ≤ ζ < γβ and ζ = t[δ1,...,δn,ǫ1,...,ǫm], where δ1 <...<δn <γα, γβ <ǫ1,...,ǫm. This means that

Lκ ∃δ1,...,δn <γα γα ≤ t[δ1,...,δn,ǫ1,...,ǫm] <γβ.

−1 Using the information we have for Fα = iα from lemma 4.35, we get that

Lκ ∃δ1,...,δn <γγ ≤ t[δ1,...,δn,ǫ1,...,ǫm] <γβ which is a contradiction since t[δ1,...,δn,ǫ1,...,ǫm] ∈ Mβ, thus there should be an ordinal ζ ∈ Mβ such that γ ≤ ζ<γβ.

Lemma 4.37. Γ= {γα : α<ω1} is a set of indiscernibles for Lκ.

Proof. Suppose that α1 <...<αn, β1 <...<βn and max{αn,βn} < δ1 < ...<δn are increasing sequences of elements of Γ. Using the above lemma 2n times, we get that for every formula φ:

Lκ φ[α1,...,αn] ↔ Lκ φ[δ1,...,δn] and Lκ φ[β1,...,βn] ↔ Lκ φ[δ1,...,δn].

This means that Γ is a set of indiscernibles for Lκ.

Step 3 The proof is completed using the following lemma:

Lemma 4.38. If there is a model Lλ which has a set of indiscernibles of order κ>ω, then there is a limit ordinal γ and a set I ⊂ γ, of order type κ, such that (Lγ, I) is an E-M remarkable set. Proof. See [8]. By the preceding lemma we see that there is a well-founded remarkable E-M set, hence 0♯ exists.

In the previous chapter we saw that the existence of an nontrivial elementary embedding of V is equivalent to the existence of a measurable cardinal. Theorem 4.30 is important because it implies that the existence of elementary embeddings of inner models of V can yield the truth of weaker large cardinal axioms. Using the methods described in this section one could prove 4.4. ELEMENTARY EMBEDDINGS OF L 67 relative results for elementary embeddings of the models L(X), X ⊂ ω and ♯ X = {φ : (Lℵω (X), ∈,X) φ[ℵ1,..., ℵn]}. Those methods can be extended further more using the technique of ultrapowers and lead to a ﬁner analysis of the structure of large cardinals. What we wish to study though, is how strong large cardinals we can get by extending the embeddings of V , thus we will not proceed further more to this subject. As we will see in the next chapter, assuming the existence of a nontrivial elementary embedding j : V ≺ M, the closer M is to V the stronger large cardinals we get. 68 CHAPTER 4. 0♯ AND ELEMENTARY EMBEDDINGS OF L Chapter 5

Stronger embeddings of V

The strongest large cardinals we have met up to now are the measurable cardinals. In terms of elementary embeddings, the existence of such cardinals is equivalent to the existence of an elementary embedding j : V ≺ M, where M is an inner model. In the case M = MD -D is a normal κ-complete ultraﬁlter on κ- we have seen that M 6= V , because D/∈ M, and even more M is not closed under sequences of length κ+. This means that the inner model M can be quite thin in comparison with V . In this chapter we are going to require that M is closer to V and introduce this way stronger and stronger large cardinals. Finally, we will investigate the case where M = V .

5.1 Strong, Woodin and superstrong cardinals

Our ﬁrst step in strengthening the the elementary embeddings j : V ≺ M, is done by introducing strong cardinals, Woodin cardinals and superstrong cardinals. We will deﬁne even stronger large cardinals in the next sections, but those described here have the advantage that they yield important results with relatively weaker hypotheses. They can also be described, by the notion of extenders, which are roughly sets of measures which approximates the corresponding embedding j : V ≺ M.

Strong cardinals Deﬁnition 5.1.

(i) A cardinal κ is called α-strong, if there is an elementary embedding j : V ≺ M, with critical point κ, such that α < j(κ) and Vκ+α ⊂ M.

(ii) A cardinal κ is called strong, if it is α-strong for every α ∈ On.

The following theorem contains some of the interesting consequences of the existence of α-strong or strong cardinals:

69 70 CHAPTER 5. STRONGER EMBEDDINGS OF V

Theorem 5.2. (i) If there is a strong cardinal, then for every set A V 6= L(A). (ii) For every α-strong cardinal κ, if GCH holds for every β<κ, then it also holds for every β<κ + α.

(iii) If κ is strong, then Vκ ≺2 V . Theorem 5.3. Suppose κ is 2-strong. Then, there is a normal ultraﬁlter D over κ such that {λ<κ : measurable(λ)} ∈ D. Therefore strong cardinals are stronger than measurable cardinals. Actually, the measurability of a cardinal is equivalent to its 1-strongness. Another interesting result, is the fact that the existence of a strong cardinal does not imply the existence of greater large cardinals. Theorem 5.4. cons(ZF C+∃κ strong(κ)) → cons(ZF C+∃κ strong(κ)+ 6 ∃λ > κ inaccessible(λ)).

Woodin cardinals Deﬁnition 5.5. A cardinal κ is called a Woodin cardinal, if for every f ∈κ κ, there is an α<κ and a j : V ≺ M, such that: (i) f ′′α ⊂ α; (ii) the critical point of j is α;

(iii) Vj(f)(α) ⊂ M. The great importance of Woodin’s cardinals springs from their consequences in descriptive set theory. We mention the axiom of determinacy (AD), without giving a deﬁnition because we did so, we should also state several results which reveal its importance in descriptive set theory. Theorem 5.6. The following theories are equiconsistent: (i) ZF C +“there are infinitely many Woodin cardinals”. (ii) ZF + AD. Theorem 5.7. The existence of inﬁnitely many Woodin cardinals and of a measurable cardinal above them, yields ADL(R). The following theorem, provides us with a characterization of Wooding cardinals. On the same time, it proves that cons(ZF C + ∃κ W oodin(κ) → cons(ZF C + ∃κ strong(κ) and that the existence of a Woodin cardinal, implies the existence of all α-strong cardinals, α<κ. Thus Woodin cardinals have greater consistency strength than strong cardinals. 5.1. STRONG, WOODIN AND SUPERSTRONG CARDINALS 71

Theorem 5.8. Let κ be a Woodin cardinal. Then, the set {α<κ : ∀β < κ β − strong(κ)} is stationary in κ.

One additional information we get from theorem 5.8, is that every Woodin cardinal κ is κ-Mahlo. On the contrary, it can not be weakly compact, as we 1 will see that the property of being a Woodin cardinal is Π1.

Superstrong cardinals Deﬁnition 5.9. A cardinal κ is called superstrong, if there is a j : V ≺ M with critical point κ, and such that Vj(κ) ⊂ M. If a cardinal is superstrong, then it is Woodin and there are many Woodin cardinals below it. In particular:

Theorem 5.10. Let κ be superstrong. Then κ is Woodin and there is a normal ultraﬁlter D on κ such that

{λ<κ : W oodin(λ)} ∈ D.

Extenders An extender is a set of measures, deﬁned in a similar way we deﬁned a measure from a nontrivial embedding j : V ≺ M.

Deﬁnition 5.11. Suppose that the critical point of j : V ≺ M is κ and κ ≤ γ ≤ j(κ).

<ω <ω (i) For every α ∈ [γ] , Eα is a measure on [κ] such that

X ∈ Eα ↔ α ∈ j(X).

<ω (ii) The (κ,γ)-extender derived from j is the set E = {Eα : α ∈ [β] }.

Every element E[α] of an extender (κ,γ), defines an ultrapower MEα of V . <ω It is not hard to find embeddings iα,β : MEα ≺ MEβ , for all α, β ∈ [γ] , such <ω thatα ⊂ β. The collection {MEα : α ∈ [γ] } along with the embeddings iα,β, forms a directed system of models. Thus there is a model ME, such that every model Eα is elementary embedded in ME and ME is the union of the images of Eα through those elementary embeddings. This idea is similar to that of iterated ultrapowers. By studying the properties of extenders we can conclude to the fact that they can be defined in ZF C without the assumption of the existence of the embedding j. The following characterizations of strong, Woodin and superstrong cardinals, using extenders, imply that the correlated large cardinals can be defined in ZF C.

Theorem 5.12. 72 CHAPTER 5. STRONGER EMBEDDINGS OF V

+ (i) κ is α-strong iﬀ there is a (κ, |Vκ+α| )-extender E such that Vκ+α ⊂ ME and α < jE (κ). Therefore, κ is strong iﬀ there are such extenders for every α.

κ (ii) κ is Woodin iﬀ for all f ∈ κ there is an α<κ and an extender E ∈ Vκ such that:

– f ′′α ⊂ α;

– the critical point of jE is α;

– jE(f)(α)= f(α);

– VjE (f)(α) ⊂ ME.

(iii) κ is superstrong iﬀ there is a (κ,β)-extender E, β>κ, such that VjE (κ) ⊂ ME.

5.2 Strongly compact cardinals

Strongly compact cardinals have greater consistency strength than Woodin cardinals, but there is not a known connection between them and superstrong cardinals. We deﬁne them below using elementary embeddings.

Deﬁnition 5.13. Suppose κ ≤ λ.

(i) κ is λ-compact, if there is an elementary embedding j : V → M with critical point κ and such that:

X ⊂ M ∧ |X|≤ λ → ∃Y ∈ M (X ⊂ Y ∧ M |Y | < j(κ)).

(ii) κ is strongly compact if it is λ-compact for every λ ≥ κ.

Definition 5.14. For every x ∈ Pκ(X),x ˆ = {y ∈ Pκ(X): x ⊂ y}. A κ- complete ultrafilter U on Pκ(X) is called a fine measure on Pκ(X), if ∀x ∈ Pκ(X)x ˆ ∈ U.

Using complete filters, fine filters and infinitary languages, we can give the following characterizations of strongly compact cardinals:

Lemma 5.15. For every regular cardinal κ the following are equivalent:

(i) κ is strongly compact.

(ii) For every set S, every κ-complete ﬁlter on S can be extended to a κ- complete ultraﬁlter on S.

(iii) For every X, |X|≥ κ, there is a ﬁne measure on Pκ(X).

(iv) L satisﬁes the compactness theorem. 5.3. SUPERCOMPACT CARDINALS 73

Each of those definitions yields that strong cardinals can be defined in ZF C. As we have already mentioned, the consistency strength of strong cardinals is greater than that of Woodin cardinals, therefore greater than that of measurable cardinals. We will state a stronger theorem towards this direction. In order to do this we are going to define Mitchell’s order. Mitchell’s order assists us in constructing measurable cardinals with greater consistency strength and such that many measurable cardinals lie below them. For example, if D is a normal measure on κ of order greater than 0, then {λ<κ : measurable(λ)} ∈ D. Definition 5.16.

(i) Suppose κ is a measurable cardinal and D is the collection of all the normal measures on κ. Mitchell’s order, is an order on D deﬁned the following way:

(ii) It is easy to check that the above order is well-founded, thus we may define o(D) = rank<(D), for every D ∈ D. By o(κ), we denote the supremum of {o(D): D ∈ D}. Theorem 5.17. (Mitchell) If there is a strongly compact cardinal, there exists an inner model with a measurable cardinal κ, such that o(κ)= κ++. The following yields that if there is a strongly compact cardinal, then the universe is not even relatively constructible. Theorem 5.18. (Vop˘enka-Hrbá˘cek) If there is a strongly compact cardinal, then for every set A, V 6= L[A]. We finally give the following result, which relates strongly compact cardinals with the computation of the continuum function. Theorem 5.19. (Solovay) If κ is strongly compact then the singular hypothesis holds above κ.

5.3 Supercompact cardinals

Supercompact cardinals form a very important class of large cardinals since they have strong consequences on the structure of V above them. It is hard though, to approach the structure of the universe below them, because no procedure, such as approximation with extenders, is known to work for them neither has a canonical inner model been found for them yet. The consistency strength of a supercompact cardinal is greater or equal to that of a strongly compact cardinal, but it is not known if it is strictly above. Deﬁnition 5.20.

(i) Suppose κ ≤ λ. κ is λ-supercompact if there is an embedding j : V ≺ M such that: 74 CHAPTER 5. STRONGER EMBEDDINGS OF V

– the critical point of j is κ; – j(κ) > λ; – λM ⊂ M. (ii) κ is supercompact if it is λ-supercompact for all λ ≥ κ. The following characterization of λ-supercompactness implies that it can be defined in ZF C. The same of course holds for supercompactness. The characterization uses normal filters on Pκ(λ) and it works in a similar way as ultrafilters are used in the characterization of measurable cardinals.

Definition 5.21. A fine filter on Pκ(λ) is normal, if it is closed under diagonal intersections of the form

△α<λXα = {x ∈ Pκ(λ): x ∈ Xα}. α\∈x The same way as we did for normal measures, we can prove the following equivalence:

1 Lemma 5.22. A filter D over Pκ(λ) is normal iff every choice function on a D-stationary set2 X is constant on a D-stationary set. Theorem 5.23. If κ ≤ λ, then κ is λ-supercompact iff there is a normal ultrafilter over Pκ(λ). We give one more characterization of supercompactness, which introduces it as a reflection principle. Theorem 5.24. (Magidor) κ is supercompact iff for every α>κ there is a β<κ and an embedding i : Vβ ≺ Vα with critical point δ, i(δ)= κ. We present some results which -as we have already seen- are also true for strong cardinals. Theorem 5.25.

(i) If κ is 2κ-supercompact, then there is a normal ultraﬁlter D over κ such that {λ<κ : measurable(λ)} ∈ D.

(ii) If κ is γ-supercompact, then ∀α<κ 2α = α+ implies that ∀β ≤ γ 2β = β+.

(iii) If κ is supercompact, then Vκ ≺2 V . Finally, we see that supercompact cardinals have many measures and their Mitchell order is high.

1∀x ∈ X \ ∅ f(x) ∈ x. 2If F is a ﬁlter on S then X ⊂ S is F -stationary if ∀Y ∈ F X ∩ Y 6= ∅. 5.4. EXTENDIBLE CARDINALS AND VOPENKA’S˘ PRINCIPLE 75

Theorem 5.26. Let κ be supercompact, then κ (i) there are 22 normal measures on κ;

2<λ (ii) for every λ ≥ κ there are 2 normal measures on Pκ(λ); (iii) the Mitchell order of κ is (2κ)+.

5.4 Extendible cardinals and Vop˘enka’s principle

Here we see some large cardinal of even greater consistency strength.

Extendible cardinals Deﬁnition 5.27. (α 6= 0)

(i) A cardinal κ is α-extendible if there is a β and an embedding i : Vκ+α → Vβ with critical point κ and such that α

(ii) Suppose κ is |Vκ+α|-supercompact. Then, there is a normal measure D on κ such that {β<κ : α − extendible(β)} ∈ D.

(ii) If κ is α-extendible and β +2 < α, then κ is |Vκ|-supercompact. Finally, we give one important reﬂection property of extendible cardinals, out of which, many consequences of the existence of extendible cardinals can be derived.

Theorem 5.31. If κ is extendible, then Vκ ≺3 V . 76 CHAPTER 5. STRONGER EMBEDDINGS OF V

Vop˘enka’s principle In contrast with the large cardinal axioms we have considered up to now, Vop˘enka’s principle does not imply the existence of a specific large cardinal. This is because it requests the existence of several elementary embeddings. Definition 5.32. Vop˘enka’s principle (VP): Let C be a class of structures of the language L. There are A, B ∈ C such that there is an embedding i : A ≺ B. Vop˘enka’s principle is stronger than extendible cardinals: Theorem 5.33. VP implies that the class of extendible cardinals κ, carrying a normal measure D such that {λ<κ : extendible(λ)} ∈ D, is stationary on On. We can restate VP, in terms of A-supercompact and A-extendible cardinals, which are relativized versions of supercompact and extendible cardinals respectively. Definition 5.34. Let A be a class and κ a cardinal. (i) κ is A-supercompact if for all α>κ there is a β<κ and an embedding

i : (Vβ , ∈, Vβ ∩ A) ≺ (Vα, ∈, Vα ∩ A)

with critical point γ such that i(γ)= κ. (ii) κ is A-extendible if for all α>κ there is an ordinal β and an embedding

i : (Vα, ∈, Vα ∩ A) ≺ (Vβ, ∈, Vβ ∩ A)

with critical point κ and such that α

5.5 Huge cardinals and I0-I3 Huge cardinals Deﬁnition 5.36.

(i) κ is n-huge if there is an embedding j : V ≺ M with critical point κ and n such that j (κ)M ⊂ M. (ii) κ is huge if it is 1-huge. 5.5. HUGE CARDINALS AND I0-I3 77

The following equivalent form of n-hugeness, yields that it can be deﬁned in ZF C.

Theorem 5.37. A cardinal κ is n-huge iﬀ there are cardinals κ = λ0 < λ1 < ... < λn = λ and a κ-complete normal ultraﬁlter D over P (λ) such that for every i

Although huge cardinals have greater consistency strength, the existence of a huge cardinal does not imply the existence of a supecompact cardinal. Even more:

Theorem 5.38. If there are both a huge and a supercompact cardinal, then the least huge cardinal is below the least supercompact cardinal.

Theorem 5.39. cons(ZF + ∃κ huge(κ)) → cons(ZF + V P ). In particular, if κ is huge, then Vκ V P .

It is known that n-huge cardinals form an increasing in strength hierarchy.

Theorem 5.40. If κ is n +1-huge, then there is a normal measure D on κ such that {λ<κ : n − huge(λ)}.

I0-I3

The principles I0-I3 are very close to inconsistency since they require the existence of embeddings close to j : V ≺ V . We deﬁne them below and place them in the hierarchy of large cardinals.

Deﬁnition 5.41.

I0: There is an α and an i : L(Vα+1) ≺ Vα+1 with critical point strictly less than α.

I1: There is an α and an embedding i : Vα+1 ≺ Vα+1.

I2: There is an α and an embedding j : V ≺ M such that Vα ⊂ M, the critical point of j is strictly less than α and j(α)= α.

I3: There is an α and an embedding i : Vα ≺ Vα.

Theorem 5.42. I0 → I1 → I2 → I3.

Theorem 5.43. If I3 holds and the critical point of the corresponding embedding j is κ, then there is a normal measure D on κ such that {λ<κ : ∀n < ω n − huge(κ)} ∈ D. 78 CHAPTER 5. STRONGER EMBEDDINGS OF V 5.6 Kunen’s theorem

Up to now we have used elementary embeddings in order to define new large cardinal axioms. We may notice that as we try to increase the consistency strength of those axioms, we defined each time an elementary embedding closer to j : V ≺ V . In the following two we investigate whether such an embedding exists and in the case it exist we state its consequences. By a theorem due to Kunen, we see that if we extend the language of set with a function symbol j such that j is an elementary embedding and it satisfies all instances of replacement for the formulas of the extended language. On the contrary, if we do not require that j satisfies the replacement axiom we cannot get to a contradiction as in Kunen’s theorem. This situation is described by the wholeness axiom, introduced by Corazza, which is a large cardinal axiom with consistency strength between 3 n-huge cardinals and I3. We will show that there is no extension (V, ∈, j) of V such that j is a nontrivial embedding j : V ≺ V which satisfies all replacement for every formula φ of the language (∈, j). In order to achieve this, we will first define ω-Jónsson functions that will help us conclude to a contradiction.

Deﬁnition 5.44. A function f :ω x → x is called ω-J´onsson on x if y ⊂ x ∧ |y| = |x| → f ′′ωy = x.

Theorem 5.45. (Erd¨os-Hajnal) For every λ ≥ ω, there is an ω-J´onsson function over λ.

Theorem 5.46. If j : V ≺ M, then M 6= V .

Proof. Suppose M = V , the critical point of j is κ and λ = sup{jn(κ): n<ω}. We have that j(λ) = sup{jn+1(κ): n<ω} = λ. On the other hand, there is an ω-J´onsson function f over λ. j(f) is also ω- J´onsson. Let X = j′′λ. |X| = λ, thus there is a x ∈ω X such that j(f)(x)= κ. If x(n)= j(αn) for every n<ω, then x = j(y), where ∀n<ωy(n)= αn. The latter means that κ = j(f)(j(y)) = j(f(y)) which is a contradiction, as κ is the critical point of j.

The following theorem is similar to the above, and it emerges from some other proofs of 5.46.

Theorem 5.47.

(i) For any α there is no j : Vα+2 ≺ Vα+2.

(ii) If α is the least ordinal above the critical point of j, such that j(α) = α, then j′′α∈ / M.

3In fact super-n-huge which are stronger than n-huge. 5.7. THE WHOLENESS AXIOM 79 5.7 The wholeness axiom

As we have already seen, the axiom of replacement was used in order to construct n the sequence {j (κ)}n<ω. Hence, in the case we do not require that j satisfies the replacement axiom the proof is no longer valid. This situation is captured by the notion of weak definability. Definition 5.48. Let M be a model of ZF and X ⊂ M. X is weakly definable in M if (M, ∈,X) satisfies the axiom of replacement for the extended language. Definition 5.49. Wholeness Axiom (WA): There is a nontrivial embedding j; V ≺ V which is not weakly definable in V and such that for every set A, j ↾ A is a set. We call j a WA-embedding. The request for j ↾ A to be a set is necessary in order not to lose all the strength obtained by the replacement axiom. This specific part of WA places it high in the hierarchy of large cardinals. In particular: Theorem 5.50. Let j : V ≺ V be W A-embedding with critical point κ. Then, κ is n-huge for every n<ω.

Theorem 5.51. If α is the least limit ordinal for which Vα ≺ Vα, then Vα W A + I3. Therefore, cons(ZF C + I3) → cons(ZF C + W A) while cons(ZF C + W A) 6→ cons(ZF C + I3). Following the proof of Kunen’s theorem we can disprove the existence of some sets if W A holds. For example: Theorem 5.52. If j : V ≺ V is an elementary embedding with critical point κ, then the sequence κ, j(κ), jn(κ),... is not weakly definable in V and it is unbounded in On. We finish this section with an important reflection principle emerging from W A.

Theorem 5.53. If j : V → V is a W A-embedding, then ∀n<ωVjn(κ) ≺ V . 80 CHAPTER 5. STRONGER EMBEDDINGS OF V Bibliography

[1] J. L. Bell and A. B. Slomson, Models and ultraproducts: an introduction. Dover, 2006.

[2] C. C. Chang and H. J. Keisler, Model Theory. North Holland, Amster- dam, 1973.

[3] P. Corazza, The spectrum of elementary embeddings j : V → V . Annals of Pure and Applied Logic, vol. 139, pp. 327-399, 2006.

[4] P. Corazza, The wholeness axiom. Consciousness-based education: A foundation for teaching and learning in the academic disciplines, Consciousness- based education and Mathematics, vol. 5, MUM Press, 2009.

[5] K. J. Delvin, Constructibility. Springer, Berlin, 1984.

[6] C. Dimitracopoulos, Introduction to mathematical logic. Athens, 1999.

[7] F. R. Drake, Set Theory: an introduction to large cardinals. North Holland, Amsterdam, 1974.

[8] T. Jech, Set Theory. Springer, Berlin, 3rd millenium ed, rev. and ex- panded, 2002.

[9] R. B. Jensen, Measurable cardinals and the GCH. Axiomatic Set Theory. AMS, Proccedings of Symposia in Pure Mathematics vol. 13, part 2, 1974.

[10] A. Kanamori, The Higher Inﬁnite. Springer, Berlin, 2nd edition, 2004.

[11] K. Kunen, Set Theory: an introduction to independence proofs. North Holland, Amsterdam, 1980.

[12] A. Levi, Basic set theory. Springer, Berlin, 1979.

[13] P. Maddy, Believing the axioms, I &II. The Journal of Symbolic Logic, vol.53, no. 2-3, pp. 481-511 & 736-764, 1988.

[14] Y. N. Moschovakis, Descriptive Set Theory. AMS, Mathematical sur- veys and monographs; v.155, 2nd edition, 2009.

81 82 BIBLIOGRAPHY

[15] Y. N. Moschovakis, Notes on Set Theory. Springer, Berlin, 2nd edition, 2006. [16] G. Koumoulis, S. Negrepontis, Measure Theory. Simmetria, Athens 1988. [17] W. N. Reinhardt, Remarks on reflection principles, large cardinals and elementary embeddings. Axiomatic Set Theory. AMS, Proccedings of Sym- posia in Pure Mathematics vol. 13, part 2, pp. 189-205, 1974. [18] R. M. Solovay, W. N. Reinhardt, A. Kanamori, Strong Axioms of Infinity and Elementary Embeddings. Annals of Mathematical Logic , vol.13, pp.73- 116, 1978. [19] R. M. Smullian, Gödel’s incompleteness theorems. Oxford Univesity press, 1992. [20] J. R. Steel, What is a Woodin cardinal?. Notices of the American Mathe- matical Society, 54 (9): 11467, 2007. [21] A. Tzouvaras, Elements of Mathematical Logic. Ziti, Thessaloniki, 1986. [22] H. W. Woodin, The continuum hypothesis, part I-II. Notices of the Amer- ican Mathematical Society, 48, 2001. Index

A-supercompact, 76 extendible, 75 A-extendible, 76 M-ultrafilter, 63 filter on S, 18 α-extendible, 75 fine measure, 72 α-strong, 69 κ-complete measure, 63 huge, 76 λ-supercompact, 73 ω-Jónsson, 78 inaccessible cardinal, 26 σ-additive, 37 indiscernibles, 53 n-huge, 76 inner model, 14 absolute, 7 Lévy’s hierarchy, 7 absolute for M, 7 absolute for a theory, 7 Mahlo cardinal, 31 absolute for M, N, 7 Mitchell’s order, 73 atom, 38 atomless measure, 38 nonprincipal measure, 63 normal measure, 48 canonical embedding, 19, 44 canonical Skolem function, 55 partition, 33 canonical well-ordering, 15 closed unbounded set, 30 Ramsey cardinal, 35 cofinal function, 24 real-valued measurable cardinal, 39 cofinality, 24 regressive function, 31 constructible sets, 4 regular cardinal, 26 critical point, 43 cumulative hierarchy, 3 set constructible from A, 16 set-like relation, 5 definable in M from I, 4 Singular Cardinal Hypothesis, 25 definable over M, 4 Skolem functions, 17 directed system of models, 71 Skolem hull, 17 stationary set, 31 E-extension, 5 strong, 69 E-M set, 55 strong limit cardinal, 26 elementary chain, 18 supercompact, 74 elementary embedding, 17 superstrong, 71 elementary substructure, 7 extender, 71 two-valued measure, 38

83 84 INDEX

Ulam matrix, 39 ultraﬁlter, 18 ultrapower, 19 unbounded set, 30 von Neumann’s hierarchy, 3 Vop˘enka’s principle, 76

WA-embedding, 79 weakly compact cardinal, 33 weakly deﬁnable, 79 weakly inaccessible cardinal, 32 weakly Mahlo cardinal, 32 well-founded relation, 5 Wholeness Axiom, 79 Woodin cardinal, 70